[{"data":1,"prerenderedAt":1047},["ShallowReactive",2],{"layer:prime-and-composite:extend":3},{"layer":4,"contentHash":1023,"dependencyHashes":1024,"approval":1041,"releaseId":1046},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":37,"sourceIds":1018,"reviewStatus":1019,"authoring":1020},1,"prime-and-composite","en","extend","Primes in the wild: cicadas, codes and unsolved puzzles","From insect life cycles and online banking to record primes, perfect numbers and problems nobody has solved","Take primes into the world: prime cicada cycles, the prime-based codes behind online payments, Mersenne primes and perfect numbers, Goldbach’s and the twin prime conjectures, Indian mathematicians, other number bases, olympiad puzzles and projects.",[13,14,15,16,17],"Use LCMs to explain why prime life cycles may help periodical cicadas.","Explain the one-way idea behind prime-based cryptography and its limits.","Connect Mersenne primes to perfect numbers and classify numbers as perfect, abundant or deficient.","Describe Goldbach’s conjecture, the twin prime conjecture and other open problems accurately.","Solve olympiad-style problems using remainders, factor counts and prime factorisations.",60,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Time","≈ 60 minutes, plus projects",{"label":29,"value":30},"Chapters","11",{"label":32,"value":33},"Labs","Perfect sort, famous trees, famous match, grand sieve",{"label":35,"value":36},"Open problems","Twin primes, Goldbach, odd perfect numbers",[38,42,48,54,57,107,111,116,132,138,143,146,171,176,187,192,197,200,238,242,268,273,276,351,356,361,364,406,410,422,427,430,457,462,509,514,519,522,551,556,561,571,581,591,600,609,621,642,662,696,701,751,767,772,775,779,783,788,793,797,855,985,989,1003],{"id":39,"type":40,"markdown":41},"intro-extend","prose","You now know what primes are, how to find them, and why they behave as they do. This layer takes them out into the world and to the edge of what anyone knows.\n\nYou will see why insects may have evolved prime life cycles, how enormous primes protect the payments you make on a phone, how volunteers with home computers hunt record-breaking primes, and which simple-sounding questions about primes have defeated the greatest mathematicians for centuries. There are olympiad-style puzzles to crack and projects to try. Pick what interests you: this layer is a buffet, not a set menu.",{"id":43,"type":44,"variant":45,"title":46,"markdown":47},"how-to-use-e","callout","observation","How to use this layer","Each chapter stands on its own. Puzzles have worked solutions, but give each one at least ten minutes of your own thinking first. A calculator is fine for the big multiplications; the ideas are what matter.",{"id":49,"type":50,"title":51,"eyebrow":52,"navLabel":53},"ch1","chapter","Why cicadas count in primes","Chapter 01","1 Prime cicadas",{"id":55,"type":40,"markdown":56},"cicada-e","**Periodical cicadas** of North America spend almost their whole lives underground as nymphs, feeding on sap from tree roots. Then, in one spring, millions crawl out at once, climb trees, sing, mate, lay eggs and die within a few weeks. Some broods do this every **13 years**, others every **17 years**.\n\nWhy primes? One leading idea is about **predators**. Suppose a predator population booms every few years. A cicada that emerges every 12 years would meet predators on 2-, 3-, 4- and 6-year cycles **at every single emergence**, because 2, 3, 4 and 6 are all factors of 12. A 13-year cicada shares no factor with any of these, so it lines up with each predator's boom far less often. The table shows how many years pass between meetings (the **lowest common multiple** of the two cycles).",{"id":58,"type":59,"caption":60,"columns":61,"rows":68},"table-cicada","table","Years between a cicada emergence coinciding with a predator boom (the LCM of the two cycles)",[62,63,64,65,66,67],"Cicada cycle","Predator every 2 yrs","every 3 yrs","every 4 yrs","every 5 yrs","every 6 yrs",[69,73,80,86,90,95,102],[70,71,71,71,72,71],"12 years","12","60",[74,75,76,77,78,79],"13 years (prime)","26","39","52","65","78",[81,82,83,84,85,83],"14 years","14","42","28","70",[87,88,89,72,89,88],"15 years","30","15",[91,92,93,92,94,93],"16 years","16","48","80",[96,97,98,99,100,101],"17 years (prime)","34","51","68","85","102",[103,104,104,105,106,104],"18 years","18","36","90",{"id":108,"type":44,"variant":45,"title":109,"markdown":110},"obs-cicada","What the table shows","The rows for **13** and **17**, the primes, have the biggest numbers in every column: those cicadas meet each predator as rarely as possible. The 12-year and 18-year rows meet most predators every time they emerge.\n\nA second idea: prime cycles also stop different broods from meeting and interbreeding. A 13-year and a 17-year brood in the same place emerge together only once every 13 × 17 = **221 years**. In 2024 two neighbouring broods (Brood XIII, 17-year, and Brood XIX, 13-year) did emerge in the same year, for the first time since 1803, and their ranges touched in parts of Illinois.",{"id":112,"type":44,"variant":113,"title":114,"markdown":115},"model-limit-cicada","model_limit","A hypothesis, not a settled fact","Scientists still debate why the cycles are prime. The predator idea is appealing, but it has been hard to find real predators whose numbers boom on regular 2-to-6-year cycles. Other explanations involve ice-age climate and avoiding hybrid broods. Mathematics shows that prime cycles *would* help; biology has to show whether that is what really happened.",{"id":117,"type":118,"itemId":119,"prompt":120,"check":121,"hints":126,"feedback":129},"prac-cicada","practice","prime-and-composite.extend-cicada-lcm","A 17-year cicada and a predator with a **4-year** cycle both appear this year. In how many years will they next appear together?",{"kind":122,"answer":123,"tolerance":124,"unit":125},"number",68,0,"years",[127,128],"Find the lowest common multiple of 17 and 4.","17 and 4 are co-prime, so the LCM is their product.",{"correct":130,"incorrect":131},"Yes: 17 and 4 are co-prime, so LCM = 17 × 4 = 68 years.","The next meeting is at the LCM of 17 and 4. They are co-prime, so it is 17 × 4 = 68 years.",{"id":133,"type":134,"conceptId":135,"relation":136,"explanation":137},"conn-hcf-e","connection","hcf-and-lcm","applied_in","The cicada puzzle is an LCM problem: co-prime cycles have the largest possible LCM, their product.",{"id":139,"type":50,"title":140,"eyebrow":141,"navLabel":142},"ch2","Primes that guard your money","Chapter 02","2 Primes and security",{"id":144,"type":40,"markdown":145},"crypto-1","Every time someone pays with a UPI app, logs in to net banking or buys something online, their phone and the bank's computer must agree on secret keys over the open internet, where anyone might be listening. The mathematics that makes this possible is called **public-key cryptography**, and one of its most famous forms rests on prime numbers.\n\nThe key idea is a **one-way street**:\n\n- **Multiplying** two primes is easy. A computer multiplies two 300-digit primes in a tiny fraction of a second. Even by hand, 101 × 103 = 10,403 takes a minute.\n- **Undoing** it, **factorising** the product to find the two primes, is extremely hard when the primes are huge. Given only 10,403, you have to search for a factor (you would need to try primes up to 101). For a product with 600 digits, no known method on today's computers finishes in any reasonable time.",{"id":147,"type":148,"title":149,"items":150},"steps-rsa","steps","How a prime-based lock (RSA) works, in outline",[151,155,159,163,167],{"title":152,"tag":153,"text":154},"Pick two huge primes","secret","The bank’s computer secretly chooses two random primes, each hundreds of digits long, p and q.",{"title":156,"tag":157,"text":158},"Multiply them","public","It publishes n = p × q as part of its public key. Anyone can see n.",{"title":160,"tag":161,"text":162},"Anyone can lock","encrypt","Your phone uses n (and another public number) to scramble a message. Scrambling needs only n.",{"title":164,"tag":165,"text":166},"Only the owner can unlock","decrypt","Unscrambling needs a secret number that is easy to work out from p and q, but not from n alone.",{"title":168,"tag":169,"text":170},"Security","hard problem","To break the lock, a spy must factorise n. With primes this big, that is far beyond today’s computers.",{"id":172,"type":44,"variant":173,"title":174,"markdown":175},"nuance-crypto","nuance","What is really inside your phone","The RSA method (named after Rivest, Shamir and Adleman, who published it in 1977) is only one of several tools. Modern secure connections often use **elliptic-curve cryptography**, which is also built on number theory but not directly on factorising. Banks and payment networks, including those behind UPI, combine several methods and change them over time; the exact details are not public and differ between systems.\n\nWhat stays the same is the principle: security depends on mathematical problems that are easy one way and very hard the other way. Researchers are now designing **post-quantum** methods, because a large enough quantum computer could factorise huge numbers quickly.",{"id":177,"type":178,"title":179,"problem":180,"steps":181},"we-factor-hard","worked_example","Feel the one-way street","You are told that 9,991 is the product of two primes. Find them.",[182,183,184,185,186],"Trial division would mean testing primes up to √9,991, which is almost 100. That is 25 primes to try.","A clever shortcut: 9,991 = 10,000 − 9 = 100² − 3².","A difference of two squares factorises: a² − b² = (a − b) × (a + b). (Check with small numbers: 5² − 3² = 16 = 2 × 8.)","So 9,991 = (100 − 3) × (100 + 3) = **97 × 103**.","Real cryptography chooses primes so that no shortcut like this works. Then the only options are searches that take longer than the age of the universe.",{"id":188,"type":44,"variant":189,"title":190,"markdown":191},"ex-rsa-250","example","How big is \"too big to crack\"?","In 2020 a team of researchers used about 2,700 computer-years to factorise **RSA-250**, a 250-digit number posted as a public challenge. In September 2026 the next challenge number, the 260-digit **RSA-260**, fell too, after roughly 4,900 days of graphics-processor time; the method was the same, only the machines were faster. Keys used for real security today are typically 2,048 bits long, which is about 617 decimal digits, still far beyond those records.",{"id":193,"type":50,"title":194,"eyebrow":195,"navLabel":196},"ch3","Mersenne primes and the hunt for record primes","Chapter 03","3 Record primes",{"id":198,"type":40,"markdown":199},"mersenne","Numbers of the form **2ⁿ − 1** (multiply 2 by itself n times, then subtract 1) are called **Mersenne numbers**, after the French friar **Marin Mersenne**, who studied them in the 1600s. When a Mersenne number is prime, it is a **Mersenne prime**: 3, 7, 31, 127, 8,191, …\n\nTwo facts guide the search:\n\n- If n is composite, 2ⁿ − 1 is composite too. For example 2⁴ − 1 = 15 = 3 × 5 and 2⁶ − 1 = 63 = 7 × 9. So only **prime** n are worth testing.\n- But a prime n is not enough: 2¹¹ − 1 = 2,047 = 23 × 89.\n\nFor Mersenne numbers there is a very fast special test, the **Lucas–Lehmer test**. That is why nearly every record prime of the last 70 years has been a Mersenne prime.",{"id":201,"type":59,"caption":202,"columns":203,"rows":208},"table-mersenne","The first Mersenne numbers 2ᵖ − 1 with p prime",[204,205,206,207],"p","2ᵖ − 1","Prime?","Factors if composite",[209,214,216,219,221,225,228,231,234],[210,211,212,213],"2","3","Yes","—",[211,215,212,213],"7",[217,218,212,213],"5","31",[215,220,212,213],"127",[30,222,223,224],"2,047","No","23 × 89",[226,227,212,213],"13","8,191",[229,230,212,213],"17","131,071",[232,233,212,213],"19","524,287",[235,236,223,237],"23","8,388,607","47 × 178481",{"id":239,"type":44,"variant":189,"title":240,"markdown":241},"ex-gimps","Volunteers hunting giants","The **Great Internet Mersenne Prime Search (GIMPS)**, started in 1996, lets anyone run free software that tests Mersenne numbers when their computer is idle. It has found every record prime since then.\n\nIn October 2024 a GIMPS participant, Luke Durant, found **2¹³⁶²⁷⁹⁸⁴¹ − 1**, a prime with **41,024,320 digits**, using graphics processors rented in the cloud. Printed in ordinary type it would fill thousands of pages. It is the 52nd Mersenne prime known, although that position is provisional: not every smaller candidate has been tested yet. (Check the GIMPS site for any newer record.)",{"id":243,"type":118,"itemId":244,"prompt":245,"check":246,"hints":262,"feedback":265},"prac-mersenne","prime-and-composite.extend-mersenne-composite-n","Without calculating, why must 2¹⁵ − 1 be composite?",{"kind":247,"options":248,"correct":261},"choice",[249,252,255,258],{"id":250,"label":251},"a","Because 15 is odd",{"id":253,"label":254},"b","Because 15 is composite, and 2ⁿ − 1 is composite whenever n is",{"id":256,"label":257},"c","Because it ends in 7",{"id":259,"label":260},"d","It is actually prime",[253],[263,264],"Look at the Mersenne facts above.","15 = 3 × 5. Is 2³ − 1 = 7 a factor?",{"correct":266,"incorrect":267},"Right. 15 = 3 × 5, so 2¹⁵ − 1 is divisible by 2³ − 1 = 7 and by 2⁵ − 1 = 31. Indeed 32,767 = 7 × 31 × 151.","When n is composite, 2ⁿ − 1 is composite. Here 15 = 3 × 5, and 2¹⁵ − 1 = 32,767 = 7 × 31 × 151.",{"id":269,"type":50,"title":270,"eyebrow":271,"navLabel":272},"ch4","Perfect numbers","Chapter 04","4 Perfect numbers",{"id":274,"type":40,"markdown":275},"perfect","Add up the **proper factors** of 6 (all its factors except 6 itself): 1 + 2 + 3 = **6**. The number equals the sum of its proper factors. The ancient Greeks called such numbers **perfect**. The next one is 28: 1 + 2 + 4 + 7 + 14 = 28. Then come **496** and **8,128**, and then nothing until 33,550,336.\n\nNumbers whose proper factors add up to **more** than the number, like 12 (1 + 2 + 3 + 4 + 6 = 16), are called **abundant**. Numbers whose proper factors add up to **less**, like 8 (1 + 2 + 4 = 7) and every prime (just 1), are called **deficient**.\n\n**The Mersenne link.** Euclid proved that whenever 2ᵖ − 1 is prime, the number 2ᵖ⁻¹ × (2ᵖ − 1) is perfect:\n\n- p = 2: 2 × 3 = 6\n- p = 3: 4 × 7 = 28\n- p = 5: 16 × 31 = 496\n- p = 7: 64 × 127 = 8,128\n\nAbout 2,000 years later, Euler proved that **every even perfect number** has this form. So each new Mersenne prime gives a new perfect number, and there are exactly as many even perfect numbers known as Mersenne primes.",{"id":277,"type":278,"component":279,"componentVersion":5,"config":280,"objective":345,"textAlternative":346,"help":347},"lab-perfect-sort","interactive","sort-game",{"prompt":281,"bins":282,"items":291,"seconds":124},"Add each number’s proper factors (all factors except the number). Is it perfect, abundant or deficient?",[283,285,288],{"id":274,"label":284},"Perfect (sum = n)",{"id":286,"label":287},"abundant","Abundant (sum more)",{"id":289,"label":290},"deficient","Deficient (sum less)",[292,296,300,303,306,309,312,316,320,323,326,329,333,337,341],{"id":293,"label":294,"bin":274,"why":295},"n6","6","Proper factors add to 6, exactly the number.",{"id":297,"label":298,"bin":289,"why":299},"n8","8","Proper factors add to 7, less than the number.",{"id":301,"label":71,"bin":286,"why":302},"n12","Proper factors add to 16, more than the number.",{"id":304,"label":226,"bin":289,"why":305},"n13","Proper factors add to 1, less than the number.",{"id":307,"label":89,"bin":289,"why":308},"n15","Proper factors add to 9, less than the number.",{"id":310,"label":104,"bin":286,"why":311},"n18","Proper factors add to 21, more than the number.",{"id":313,"label":314,"bin":286,"why":315},"n20","20","Proper factors add to 22, more than the number.",{"id":317,"label":318,"bin":289,"why":319},"n25","25","Proper factors add to 6, less than the number.",{"id":321,"label":84,"bin":274,"why":322},"n28","Proper factors add to 28, exactly the number.",{"id":324,"label":88,"bin":286,"why":325},"n30","Proper factors add to 42, more than the number.",{"id":327,"label":105,"bin":286,"why":328},"n36","Proper factors add to 55, more than the number.",{"id":330,"label":331,"bin":289,"why":332},"n45","45","Proper factors add to 33, less than the number.",{"id":334,"label":335,"bin":289,"why":336},"n49","49","Proper factors add to 8, less than the number.",{"id":338,"label":339,"bin":274,"why":340},"n496","496","Proper factors add to 496, exactly the number.",{"id":342,"label":343,"bin":286,"why":344},"n945","945","Proper factors add to 975, more than the number.","Classify numbers as perfect, abundant or deficient by adding their proper factors.","This game gives 15 numbers and three bins. For each number, add all its factors except the number itself.\n\n- 6: proper factors add to 6 → **perfect**\n- 8: proper factors add to 7 → **deficient**\n- 12: proper factors add to 16 → **abundant**\n- 13: proper factors add to 1 → **deficient**\n- 15: proper factors add to 9 → **deficient**\n- 18: proper factors add to 21 → **abundant**\n- 20: proper factors add to 22 → **abundant**\n- 25: proper factors add to 6 → **deficient**\n- 28: proper factors add to 28 → **perfect**\n- 30: proper factors add to 42 → **abundant**\n- 36: proper factors add to 55 → **abundant**\n- 45: proper factors add to 33 → **deficient**\n- 49: proper factors add to 8 → **deficient**\n- 496: proper factors add to 496 → **perfect**\n- 945: proper factors add to 975 → **abundant**\n\nAll primes are deficient (their only proper factor is 1). 945 is the smallest odd abundant number.",{"hints":348},[349,350],"List factor pairs, then drop the number itself before adding.","Primes are always deficient.",{"id":352,"type":44,"variant":353,"title":354,"markdown":355},"question-odd-perfect","question","Is there an odd perfect number?","Every perfect number anyone has found is even. Nobody has ever found an odd one, and nobody has proved that none exists. Computer searches show that an odd perfect number, if it exists, would have to be larger than 10¹⁵⁰⁰, a number with more than 1,500 digits. This may be the oldest unsolved problem in mathematics.",{"id":357,"type":50,"title":358,"eyebrow":359,"navLabel":360},"ch5","Goldbach’s conjecture","Chapter 05","5 Goldbach",{"id":362,"type":40,"markdown":363},"goldbach","In 1742 **Christian Goldbach** and **Leonhard Euler** exchanged letters about this idea:\n\n> **Every even number greater than 2 is the sum of two primes.**\n\n4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, … and 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53.\n\nComputers have checked every even number up to **4 × 10¹⁸** (four million million million). It is true every time. Yet nobody has proved it must be true for **all** even numbers. It is called **Goldbach’s conjecture**, and it is one of the most famous unsolved problems in mathematics.\n\nA close cousin, the **weak Goldbach conjecture** (every odd number greater than 5 is the sum of three primes), was proved in 2013 by the Peruvian mathematician **Harald Helfgott**. Experts accept his proof, though the full written version is still being prepared for publication.",{"id":365,"type":59,"caption":366,"columns":367,"rows":371},"table-goldbach","Ways to write even numbers as the sum of two primes (order ignored)",[368,369,370],"Even number","Ways","The pairs",[372,376,378,380,383,385,387,389,392,394,397,400,403],[373,374,375],"4","1","2 + 2",[294,374,377],"3 + 3",[298,374,379],"3 + 5",[381,210,382],"10","3 + 7, 5 + 5",[71,374,384],"5 + 7",[314,210,386],"3 + 17, 7 + 13",[84,210,388],"5 + 23, 11 + 17",[390,373,391],"50","3 + 47, 7 + 43, 13 + 37, 19 + 31",[72,294,393],"7 + 53, 13 + 47, 17 + 43, 19 + 41, 23 + 37, 29 + 31",[106,395,396],"9","7 + 83, 11 + 79, 17 + 73, 19 + 71, 23 + 67, 29 + 61, …",[398,211,399],"98","19 + 79, 31 + 67, 37 + 61",[401,294,402],"100","3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53",[404,232,405],"210","11 + 199, 13 + 197, 17 + 193, 19 + 191, 29 + 181, 31 + 179, …",{"id":407,"type":44,"variant":45,"title":408,"markdown":409},"obs-goldbach","More room, more ways","The number of ways tends to grow as the even numbers get bigger, although it jumps around: 98 has only 3 ways but 90 has 9. Multiples of 3 and 5, like 60, 90 and 210, tend to have extra ways. If the pattern of growth continues forever, Goldbach must be true, but \"tends to\" is not a proof.",{"id":411,"type":118,"itemId":412,"prompt":413,"check":414,"hints":416,"feedback":419},"prac-goldbach","prime-and-composite.extend-goldbach-100","In how many ways can **100** be written as the sum of two primes? (Count 3 + 97 and 97 + 3 as the same way.)",{"kind":122,"answer":415,"tolerance":124},6,[417,418],"Try each prime p up to 50 and check whether 100 − p is prime.","Start: 3 + 97.",{"correct":420,"incorrect":421},"Yes: 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. Six ways.","There are six: 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. Pairs like 7 + 93 fail because 93 = 3 × 31.",{"id":423,"type":50,"title":424,"eyebrow":425,"navLabel":426},"ch6","The twin prime conjecture and a 2013 breakthrough","Chapter 06","6 Twin prime frontier",{"id":428,"type":40,"markdown":429},"twin-frontier","Recall the **twin prime conjecture**: there are infinitely many pairs of primes that differ by 2. It is at least 150 years old in its modern form and still unproved.\n\nFor a long time mathematicians could not even prove that there are infinitely many pairs of primes differing by **any** fixed amount, whether 2 or 1,000 or a billion. Then in **2013**, **Yitang Zhang**, a little-known lecturer at the University of New Hampshire, proved that there are infinitely many pairs of primes that differ by **less than 70 million**. It was a stunning result from someone who had worked quietly on it for years.\n\nWithin months, **James Maynard** found a different method, and a large online collaboration called **Polymath** combined the ideas. The gap came down to **246**. So we now know that infinitely many pairs of primes differ by 246 or less. Getting from 246 to 2 is the remaining challenge, and experts think it needs a genuinely new idea.\n\nMeanwhile, the largest known twin primes, found in 2016, have **388,342 digits** each.",{"id":431,"type":432,"title":433,"note":434,"scale":435,"rungs":436},"ladder-gap","ladder","The proven gap: from \"no idea\" to 246","Log scale. \"Infinitely many prime pairs differ by at most this much.\" The conjecture says 2.","log",[437,441,445,449,453],{"label":438,"value":439,"display":440},"Zhang, May 2013",70000000,"70,000,000",{"label":442,"value":443,"display":444},"Polymath, 2013",4680,"4,680",{"label":446,"value":447,"display":448},"Maynard, Nov 2013",600,"600",{"label":450,"value":451,"display":452},"Polymath, 2014",246,"246",{"label":454,"value":455,"display":456},"Twin prime conjecture",2,"2 (unproved)",{"id":458,"type":50,"title":459,"eyebrow":460,"navLabel":461},"ch7","Indian minds and prime ideas","Chapter 07","7 Indian connections",{"id":463,"type":464,"title":465,"prompt":466,"options":467},"explorer-india","explorer","Indian mathematicians and prime ideas","Choose a person to see their connection to primes and factors.",[468,478,488,501],{"id":469,"label":470,"chain":471,"note":477},"ramanujan","Srinivasa Ramanujan",[472,473,474,475,476],"Kumbakonam, Tamil Nadu","Self-taught","Letter to Hardy, 1913","Cambridge","Highly composite numbers","Ramanujan (1887–1920) filled notebooks with astonishing formulas, many about how numbers split into parts. His 1915 paper on highly composite numbers studied numbers with record numbers of factors, the opposite of primes. He also worked on how primes are spread out; \"Ramanujan primes\" are named after one of his results. His birthday, 22 December, is celebrated as National Mathematics Day in India.",{"id":479,"label":480,"chain":481,"note":487},"taxicab","1,729: the taxi-cab number",[482,483,484,485,486],"Hardy’s taxi","\"A dull number?\"","1³ + 12³","9³ + 10³","7 × 13 × 19","When Hardy said his taxi number, 1,729, seemed dull, Ramanujan replied that it is the smallest number expressible as the sum of two cubes in two different ways. Its prime factorisation, 7 × 13 × 19, is three primes 6 apart. It is also a Carmichael number, a composite that fools a well-known shortcut test for primes.",{"id":489,"label":490,"chain":491,"badge":497,"note":500},"aks","The AKS test",[492,493,494,495,496],"IIT Kanpur, 2002","Agrawal, Kayal, Saxena","\"PRIMES is in P\"","Fast and always right","Gödel Prize 2006",{"text":498,"tone":499},"A landmark in computer science","yes","Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena at IIT Kanpur found the first primality test that is always correct, works for every number, and runs in \"polynomial time\" (fast, in a precise sense) without relying on any unproved conjecture. Kayal and Saxena were undergraduates when the work began. The paper won the Gödel Prize, one of the top awards in theoretical computer science.",{"id":502,"label":503,"chain":504,"note":508},"bhargava","Manjul Bhargava",[505,506,507],"Tabla player","Number theorist","Fields Medal 2014","Manjul Bhargava, a Canadian-American mathematician of Indian origin, won the Fields Medal in 2014 for work in number theory, including powerful new methods for counting whole families of number systems and curves. He often credits Sanskrit poetry and the tabla, and ancient Indian mathematicians such as Hemachandra and Pingala, for shaping how he thinks about patterns.",{"id":510,"type":44,"variant":511,"title":512,"markdown":513},"careful-1729","careful","Fooling a shortcut","Some fast primality shortcuts are based on Fermat’s little theorem. A few composite numbers, called **Carmichael numbers**, pass that shortcut for every base co-prime to them even though they are not prime. The smallest is 561 = 3 × 11 × 17, and 1,105 and 1,729 are the next two. That is why real software uses stronger tests.",{"id":515,"type":50,"title":516,"eyebrow":517,"navLabel":518},"ch8","Primes in other number systems","Chapter 08","8 Other bases",{"id":520,"type":40,"markdown":521},"bases","Our numbers are written in **base ten**: each place is worth ten times the one to its right. Other bases are possible. Computers use **base two** (binary), and many old measuring systems used twelves and sixties.\n\n**Being prime does not depend on the base.** Thirteen objects cannot be arranged in a rectangle, whether you write the number as 13 (base ten), 1101 (base two) or 11 (base twelve). Primality is about the **quantity**, not the way it is written.\n\n**Divisibility rules do depend on the base.** The digit-sum test for 3 and 9 works because ten is one more than nine. In base twelve, the digit-sum test works for **eleven** (and its factors), because twelve is one more than eleven; and the last-digit test works for 2, 3, 4 and 6, because they all divide twelve. In binary, a number is even exactly when its last digit is 0, just as in base ten a number is a multiple of 10 when it ends in 0.",{"id":523,"type":59,"caption":524,"columns":525,"rows":529},"table-bases","The same numbers written in three bases",[526,527,528,206],"Number (base ten)","Base two","Base twelve (A = ten, B = eleven)",[530,532,534,536,539,542,544,548],[215,531,215,212],"111",[71,533,381,223],"1100",[226,535,30,212],"1101",[235,537,538,212],"10111","1B",[218,540,541,212],"11111","27",[72,543,390,223],"111100",[545,546,547,212],"97","1100001","81",[549,550,401,223],"144","10010000",{"id":552,"type":44,"variant":553,"title":554,"markdown":555},"aha-bases","aha","Why ten is a slightly awkward base","Ten has factors 1, 2, 5 and 10, so base ten has last-digit tests only for 2, 5 and 10. Twelve has factors 1, 2, 3, 4, 6 and 12, which gives easy tests for more numbers. That is one reason dozens, 12-inch feet and 60-minute hours have survived for so long.",{"id":557,"type":50,"title":558,"eyebrow":559,"navLabel":560},"ch9","Olympiad-style puzzles","Chapter 09","9 Puzzles",{"id":562,"type":178,"title":563,"problem":564,"steps":565},"we-olymp-1","Three primes in disguise","Find every prime p for which p, p + 10 and p + 14 are all prime.",[566,567,568,569,570],"Look at remainders on division by 3. 10 leaves remainder 1 and 14 leaves remainder 2.","If p leaves remainder 0, then p is a multiple of 3, so p = 3. Check: 3, 13, 17 are all prime. ✓","If p leaves remainder 1, then p + 14 leaves remainder 1 + 2 = 3, i.e. 0: p + 14 is a multiple of 3 bigger than 3. ✗","If p leaves remainder 2, then p + 10 leaves remainder 2 + 1 = 3, i.e. 0: p + 10 is a multiple of 3. ✗","So the **only** answer is **p = 3**.",{"id":572,"type":178,"title":573,"problem":574,"steps":575},"we-olymp-2","Squares of primes, minus 1","Show that if p is a prime bigger than 3, then p × p − 1 is always divisible by 24.",[576,577,578,579,580],"p × p − 1 = (p − 1) × (p + 1): the numbers just below and just above p.","p is odd, so p − 1 and p + 1 are consecutive **even** numbers. One of any two consecutive even numbers is a multiple of 4, so their product has at least 2 × 4 = 8 as a factor.","Of the three consecutive numbers p − 1, p, p + 1, one is a multiple of 3. It is not p (a prime bigger than 3), so it is p − 1 or p + 1.","So the product is divisible by 8 and by 3, which are co-prime, so it is divisible by **24**.","Check: 5 × 5 − 1 = 24; 7 × 7 − 1 = 48 = 2 × 24; 11 × 11 − 1 = 120 = 5 × 24. ✓",{"id":582,"type":178,"title":583,"problem":584,"steps":585},"we-olymp-3","Zeros at the end of 100!","100! (read \"100 factorial\") means 1 × 2 × 3 × … × 100. How many zeros does it end with?",[586,587,588,589,590],"Each final zero comes from a factor 10 = 2 × 5. There are far more 2s than 5s in the product, so count the 5s.","Multiples of 5 up to 100: 100 ÷ 5 = 20 of them, each giving at least one 5.","Multiples of 25 (25, 50, 75, 100) give a **second** 5 each: 4 more.","Total: 20 + 4 = **24 zeros**.","This works only because factorisation into primes is unique: the number of 5s is a definite fact about 100!.",{"id":592,"type":178,"title":593,"problem":594,"steps":595},"we-olymp-4","The 100 lockers","A school corridor has 100 closed lockers. Student 1 opens every locker. Student 2 toggles (opens if closed, closes if open) every 2nd locker. Student 3 toggles every 3rd, and so on up to student 100. Which lockers end up open?",[596,597,598,599],"Locker n is toggled once for each factor of n. For example, locker 12 is toggled by students 1, 2, 3, 4, 6 and 12.","A locker ends **open** if it is toggled an **odd** number of times.","Only perfect squares have an odd number of factors (their square-root factor has no partner).","So the open lockers are **1, 4, 9, 16, 25, 36, 49, 64, 81 and 100**: ten lockers.",{"id":601,"type":178,"title":602,"problem":603,"steps":604},"we-olymp-5","The pandigital prime that cannot exist","Use each of the digits 1 to 9 exactly once to make a nine-digit number. Can the number ever be prime?",[605,606,607,608],"However the digits are arranged, their sum is 1 + 2 + 3 + … + 9 = 45.","45 is a multiple of 9 (and of 3), so by the digit-sum test **every** such number is divisible by 9.","So none of the 362,880 possible arrangements is prime, without testing a single one.","What about using 1 to 8 once each? The digit sum is 36, again a multiple of 9. With 1 to 7 the sum is 28, not a multiple of 3, and the largest such prime is 7,652,413.",{"id":610,"type":118,"itemId":611,"prompt":612,"check":613,"hints":615,"feedback":618},"prac-olymp-10","prime-and-composite.extend-smallest-10-factors","What is the **smallest** number with exactly **10** factors?",{"kind":122,"answer":614,"tolerance":124},48,[616,617],"10 = 10 or 5 × 2. So the number is p⁹ or p⁴ × q.","Put the bigger power on the smaller prime.",{"correct":619,"incorrect":620},"Yes: 2⁴ × 3 = 48 has (4 + 1)(1 + 1) = 10 factors. The other shape, 2⁹ = 512, is bigger.","The two shapes are p⁹ (smallest 2⁹ = 512) and p⁴ × q (smallest 2⁴ × 3 = 48). The answer is 48.",{"id":622,"type":118,"itemId":623,"prompt":624,"check":625,"hints":636,"feedback":639},"prac-olymp-p2","prime-and-composite.extend-p-squared-plus-2","For how many primes p is **p × p + 2** also prime?",{"kind":247,"options":626,"correct":635},[627,629,631,633],{"id":250,"label":628},"None",{"id":253,"label":630},"Exactly one",{"id":256,"label":632},"Exactly two",{"id":259,"label":634},"Infinitely many",[253],[637,638],"Try p = 2, 3, 5, 7.","For p not 3, what remainder does p × p leave on division by 3?",{"correct":640,"incorrect":641},"Only p = 3 (giving 11). For any other prime, p × p leaves remainder 1 on division by 3, so p × p + 2 is a multiple of 3.","Exactly one: p = 3 gives 11. p = 2 gives 6. Every other prime leaves remainder 1 or 2 on division by 3, so p × p leaves remainder 1, and p × p + 2 is divisible by 3.",{"id":643,"type":278,"component":644,"componentVersion":5,"config":645,"objective":656,"textAlternative":657,"help":658},"lab-tree-e","factor-tree",{"numbers":646,"showIndexForm":655},[647,648,649,650,651,652,653,654],496,561,1105,1729,2047,4095,8128,9991,true,"Factorise famous numbers: perfect numbers, Carmichael numbers, a failed Mersenne number and a disguised product of two primes.","This lab gives eight famous numbers to factorise with trees.\n\n- 496 = **2⁴ × 31** (perfect: 2⁴ × (2⁵ − 1))\n- 561 = **3 × 11 × 17** (smallest Carmichael number)\n- 1,105 = **5 × 13 × 17** (second Carmichael number)\n- 1,729 = **7 × 13 × 19** (taxi-cab number, third Carmichael number)\n- 2,047 = **23 × 89** (= 2¹¹ − 1, a Mersenne number that is not prime)\n- 4,095 = **3² × 5 × 7 × 13** (= 2¹² − 1)\n- 8,128 = **2⁶ × 127** (perfect: 2⁶ × (2⁷ − 1))\n- 9,991 = **97 × 103** (= 100² − 3²)\n\nFor 9,991, spotting 10,000 − 9 saves a long search.",{"hints":659},[660,661],"Perfect numbers: keep halving until you reach an odd number.","9,991 is close to 10,000.",{"id":663,"type":278,"component":664,"componentVersion":5,"config":665,"objective":691,"textAlternative":692,"help":693},"lab-famous-match","match-pairs",{"prompt":666,"mode":667,"pairs":668},"Match each famous number with the fact that makes it famous.","memory",[669,672,674,676,679,682,684,687,689],{"a":670,"b":671},"1,729","Sum of two cubes in two ways",{"a":84,"b":673},"Second perfect number",{"a":222,"b":675},"2¹¹ − 1 = 23 × 89",{"a":677,"b":678},"561","Smallest Carmichael number",{"a":680,"b":681},"30,031","2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509",{"a":229,"b":683},"Years in a long cicada cycle",{"a":685,"b":686},"1,681","Where n × n + n + 41 first fails",{"a":343,"b":688},"Smallest odd abundant number",{"a":452,"b":690},"Proven bound on infinitely many prime gaps","Flip cards to match famous numbers from this topic with what makes them special.","This memory game hides 18 cards: 9 numbers and 9 facts.\n\n- 1,729: the smallest sum of two cubes in two ways (1³ + 12³ = 9³ + 10³).\n- 28: the second perfect number.\n- 2,047: 2¹¹ − 1 = 23 × 89, the first Mersenne number with a prime exponent that is not prime.\n- 561: the smallest Carmichael number, 3 × 11 × 17.\n- 30,031: Euclid’s product-plus-one for the first six primes, which equals 59 × 509.\n- 17: years in the longer periodical cicada cycle.\n- 1,681 = 41 × 41: where Euler’s formula n × n + n + 41 first fails (n = 40).\n- 945: the smallest odd abundant number.\n- 246: infinitely many pairs of primes differ by at most 246 (2014).",{"hints":694},[695],"Think back to the Investigate and Deepen layers for 1,681 and 30,031.",{"id":697,"type":50,"title":698,"eyebrow":699,"navLabel":700},"ch10","Projects to try","Chapter 10","10 Projects",{"id":702,"type":464,"title":703,"prompt":704,"options":705},"explorer-projects","Five prime projects","Choose a project. Each takes an afternoon or a week.",[706,715,724,733,742],{"id":707,"label":708,"chain":709,"note":714},"sieve1000","Sieve to 1,000",[710,711,712,713],"Squared paper","Rows of 20 or 30","Sieve with 2 … 31","Count 168 primes","Draw 1 to 1,000 on large squared paper. Predict the last prime you need (31, since 37 × 37 is over 1,000). Record how many new numbers each prime crosses out. Try rows of 30: why do the primes line up in just eight columns (1, 7, 11, 13, 17, 19, 23, 29 after the first row)?",{"id":716,"label":717,"chain":718,"note":723},"ulam","Ulam spiral",[719,720,721,722],"Write 1 in the centre","Spiral outwards","Colour the primes","Look for diagonals","In 1963 Stanisław Ulam doodled the numbers in a square spiral during a boring meeting and coloured the primes. Surprising diagonal lines appeared. Make your own spiral up to 400 on squared paper. Many of the diagonals follow formulas like n × n + n + 41 from the Investigate layer.",{"id":725,"label":726,"chain":727,"note":732},"cicada-model","Cicada model",[728,729,730,731],"Pick cycles","Mark emergences","Count clashes","Compare primes","On a 200-year timeline, mark the emergence years of cicadas with cycles 12 to 18, and predator booms every 2, 3, 4, 5 and 6 years. Count how many times each cicada meets a predator. Present your results as a bar chart. Do the prime cycles really win?",{"id":734,"label":735,"chain":736,"note":741},"emirps","Emirps & palindromes",[737,738,739,740],"Reverse the digits","Still prime?","Emirp!","Palindromic primes","An emirp (\"prime\" backwards) is a prime that gives a different prime when reversed: 13 ↔ 31, 17 ↔ 71, 37 ↔ 73, 79 ↔ 97. Find all emirps below 200. Then hunt palindromic primes like 101, 131 and 929. Why must every palindromic prime with an even number of digits, except 11, fail? (Test: which number do they all divide by?)",{"id":743,"label":744,"chain":745,"note":750},"goldbach-chart","Goldbach comet",[746,747,748,749],"Even n from 4 to 200","Count prime pairs","Plot the counts","See the comet","For each even number up to 200, count the ways to write it as a sum of two primes, and plot the count against the number. The points spread out like a comet’s tail, rising overall. Colour multiples of 3 differently: what do you notice?",{"id":752,"type":278,"component":753,"componentVersion":5,"config":754,"objective":761,"textAlternative":762,"help":763},"lab-grand-sieve","prime-sieve",{"max":755,"columns":756,"modes":757,"rounds":756},200,12,[758,759,760],"sieve","twins","coprime","Use the sieve, twin-prime and co-prime modes on a 12-column grid up to 200, and look for the column patterns.","This grand lab has three modes on a grid of 1 to 200 in rows of 12.\n\n- **Sieve:** with 12 columns, after the first row every prime lies in the columns under 1, 5, 7 and 11: the numbers co-prime to 12. The other eight columns contain only multiples of 2 or 3.\n- The sieve leaves 46 primes up to 200; composites that look prime include 119, 133, 143, 161, 169 and 187.\n- **Twins:** 15 pairs up to 200; after (3, 5) each pair straddles a multiple of 6, so on this grid the pairs sit in columns 5 and 7, or 11 and 1 of the next row.\n- **Co-prime:** pick 12 as the target and the co-prime numbers are exactly the four prime columns. Pick 30 and only numbers avoiding 2, 3 and 5 light up.",{"hints":764},[765,766],"Why can a prime above 3 never sit in the column under 9?","Compare with the 6-column grid from Investigate.",{"id":768,"type":50,"title":769,"eyebrow":770,"navLabel":771},"ch11","Careers, open questions and the big picture","Chapter 11","11 Open questions",{"id":773,"type":40,"markdown":774},"careers","People who work with primes and factors today include:\n\n- **Cryptographers**, who design and test the codes protecting banks, payment systems such as UPI, messaging apps and governments. India’s growing digital public infrastructure needs many of them.\n- **Cybersecurity analysts**, who make sure those codes are used correctly, since most real attacks exploit mistakes rather than the mathematics.\n- **Number theorists** at universities and research institutes such as the Indian Statistical Institute, the Tata Institute of Fundamental Research, the Chennai Mathematical Institute and the IITs.\n- **Software engineers**, who use primes in hashing, random number generators and error-correcting codes that keep data safe on phones and satellites.\n- **Quantum computing researchers**, racing to build machines that could factorise big numbers, and others racing to build codes those machines cannot break.",{"id":776,"type":44,"variant":353,"title":777,"markdown":778},"question-open","Open questions for curious learners","Nobody has solved these. Some are easy to state and have resisted the best minds for centuries.\n\n- **Twin primes:** are there infinitely many? (Known: infinitely many gaps of at most 246.)\n- **Goldbach:** is every even number above 2 a sum of two primes?\n- **Odd perfect numbers:** does even one exist?\n- **Mersenne primes:** are there infinitely many? (52 are known.)\n- **Legendre’s question:** is there always a prime between n × n and (n + 1) × (n + 1)? Check it for n up to 10 yourself.\n- **Primes of the form n × n + 1:** 2, 5, 17, 37, 101, … Do they go on forever?\n- **The Riemann Hypothesis:** a precise statement about how evenly the primes are spread, with a US $1 million prize for its solution.",{"id":780,"type":44,"variant":553,"title":781,"markdown":782},"aha-extend","The same simple idea, everywhere","Everything in this layer grows from one definition: a number with exactly two factors. From laddoo boxes to cicadas to bank security to million-dollar problems, the whole story sits on \"1 and itself\". That is the magic of mathematics: simple rules, endless depth.",{"id":784,"type":134,"conceptId":785,"relation":786,"explanation":787},"conn-patterns-e","patterns","contrasts_with","Primes famously refuse a simple formula, unlike the sequences in the patterns topic; the Ulam spiral shows glimpses of hidden order.",{"id":789,"type":134,"conceptId":790,"relation":791,"explanation":792},"conn-electricity","electricity","related_to","The secure codes that protect smart meters and the power grid’s control systems rely on the same kinds of public-key cryptography as online banking.",{"id":794,"type":134,"conceptId":795,"relation":791,"explanation":796},"conn-number-system","number-system","Divisibility rules depend on writing numbers in base ten; change the base and the rules change, though primes stay prime.",{"id":798,"type":799,"title":800,"terms":801},"gloss-extend","glossary","Words from the frontier",[802,805,809,811,815,819,823,827,830,833,837,841,845,849,852],{"term":803,"meaning":804},"cryptography","The science of keeping messages secret and secure.",{"term":806,"meaning":807,"example":808},"public-key cryptography","A system where anyone can lock a message with a public key but only the owner of a secret key can unlock it.","RSA",{"term":808,"meaning":810},"A public-key method (1977) whose security rests on how hard it is to factorise the product of two huge primes.",{"term":812,"meaning":813,"example":814},"Mersenne prime","A prime of the form 2ᵖ − 1.","7, 31, 127, 8,191",{"term":816,"meaning":817,"example":818},"perfect number","A number equal to the sum of its proper factors.","6 = 1 + 2 + 3",{"term":820,"meaning":821,"example":822},"abundant number","A number whose proper factors add up to more than itself.","12 → 16",{"term":824,"meaning":825,"example":826},"deficient number","A number whose proper factors add up to less than itself. All primes are deficient.","8 → 7",{"term":358,"meaning":828,"example":829},"The unproved claim that every even number above 2 is the sum of two primes.","28 = 5 + 23",{"term":831,"meaning":832,"example":677},"Carmichael number","A composite number that passes Fermat’s shortcut primality check for every base co-prime to it.",{"term":834,"meaning":835,"example":836},"emirp","A prime that becomes a different prime when its digits are reversed.","13 and 31",{"term":838,"meaning":839,"example":840},"palindromic prime","A prime that reads the same forwards and backwards.","131",{"term":842,"meaning":843,"example":844},"base","The number of digits a place-value system uses; each place is worth base times the one to its right.","Binary is base two.",{"term":846,"meaning":847,"example":848},"factorial","The product of all counting numbers up to n, written n!.","5! = 120",{"term":850,"meaning":851},"Lucas–Lehmer test","A fast test for whether a Mersenne number is prime.",{"term":853,"meaning":854},"periodical cicada","An insect that emerges in huge numbers every 13 or 17 years.",{"id":856,"type":857,"title":858,"questions":859},"quiz-extend","quiz","Frontier check",[860,873,886,896,908,921,933,945,956,966,975],{"itemId":861,"prompt":862,"options":863,"correct":253,"why":872},"prime-and-composite.extend-q-cicada","Why might a 13-year cycle help cicadas more than a 12-year one?",[864,866,868,870],{"id":250,"label":865},"13 is bigger",{"id":253,"label":867},"13 has no factors that match short predator cycles, so the LCMs are large",{"id":256,"label":869},"Predators cannot count past 12",{"id":259,"label":871},"13 is lucky","12 shares factors with 2-, 3-, 4- and 6-year cycles, so it meets them at every emergence. 13 is co-prime to all of them.",{"itemId":874,"prompt":875,"options":876,"correct":253,"why":885},"prime-and-composite.extend-q-rsa","The security of RSA relies mainly on…",[877,879,881,883],{"id":250,"label":878},"multiplying primes being hard",{"id":253,"label":880},"factorising a product of two huge primes being hard",{"id":256,"label":882},"there being few primes",{"id":259,"label":884},"primes being even","Multiplying is easy; recovering the two primes from their product is what is hard.",{"itemId":887,"prompt":888,"options":889,"correct":256,"why":895},"prime-and-composite.extend-q-mersenne","Which of these is a Mersenne prime?",[890,891,893,894],{"id":250,"label":89},{"id":253,"label":892},"63",{"id":256,"label":220},{"id":259,"label":222},"127 = 2⁷ − 1 is prime. 15 and 63 have composite exponents, and 2,047 = 23 × 89.",{"itemId":897,"prompt":898,"options":899,"correct":256,"why":907},"prime-and-composite.extend-q-perfect","Using Euclid’s rule 2ᵖ⁻¹ × (2ᵖ − 1), which perfect number comes from the Mersenne prime 31?",[900,902,904,905],{"id":250,"label":901},"62",{"id":253,"label":903},"248",{"id":256,"label":339},{"id":259,"label":906},"992","31 = 2⁵ − 1, so the perfect number is 2⁴ × 31 = 16 × 31 = 496.",{"itemId":909,"prompt":910,"options":911,"correct":253,"why":920},"prime-and-composite.extend-q-goldbach","What is the status of Goldbach’s conjecture?",[912,914,916,918],{"id":250,"label":913},"Proved in 1742",{"id":253,"label":915},"Checked up to 4 × 10¹⁸ but not proved",{"id":256,"label":917},"Shown to be false",{"id":259,"label":919},"Proved by Helfgott in 2013","Helfgott proved the weak (three-prime) version. The two-prime version is still open.",{"itemId":922,"prompt":923,"options":924,"correct":253,"why":932},"prime-and-composite.extend-q-zhang","What did Yitang Zhang prove in 2013?",[925,927,929,931],{"id":250,"label":926},"The twin prime conjecture",{"id":253,"label":928},"Infinitely many prime pairs differ by less than 70 million",{"id":256,"label":930},"There is a largest twin prime",{"id":259,"label":358},"His bounded-gaps result was later improved to 246.",{"itemId":934,"prompt":935,"options":936,"correct":250,"why":944},"prime-and-composite.extend-q-aks","The AKS primality test (2002) was discovered at…",[937,939,940,942],{"id":250,"label":938},"IIT Kanpur",{"id":253,"label":475},{"id":256,"label":941},"MIT",{"id":259,"label":943},"Alexandria","By Manindra Agrawal, Neeraj Kayal and Nitin Saxena.",{"itemId":946,"prompt":947,"options":948,"correct":250,"why":955},"prime-and-composite.extend-q-base","Thirteen written in base two is 1101. Is it still prime?",[949,951,953],{"id":250,"label":950},"Yes: primality is about the quantity, not the writing",{"id":253,"label":952},"No: it ends in 1",{"id":256,"label":954},"Only in base ten","Thirteen objects still cannot make a rectangle, however you write the number.",{"itemId":957,"prompt":958,"options":959,"correct":256,"why":965},"prime-and-composite.extend-q-24","For a prime p bigger than 3, p × p − 1 is always divisible by…",[960,961,962,964],{"id":250,"label":381},{"id":253,"label":92},{"id":256,"label":963},"24",{"id":259,"label":93},"(p − 1)(p + 1) has a factor 8 from the two consecutive even numbers and a factor 3. 5 × 5 − 1 = 24 shows it need not be divisible by 48.",{"itemId":967,"prompt":968,"options":969,"correct":256,"why":974},"prime-and-composite.extend-q-zeros","How many zeros does 100! end with?",[970,971,972,973],{"id":250,"label":381},{"id":253,"label":314},{"id":256,"label":963},{"id":259,"label":318},"20 multiples of 5 plus 4 extra 5s from 25, 50, 75, 100: 24 factors of 5, each paired with a 2.",{"itemId":976,"prompt":977,"options":978,"correct":250,"why":984},"prime-and-composite.extend-q-lockers","In the 100-lockers puzzle, how many lockers end open?",[979,980,981,982],{"id":250,"label":381},{"id":253,"label":318},{"id":256,"label":390},{"id":259,"label":983},"0","The perfect squares 1, 4, 9, …, 100 have an odd number of factors.",{"id":986,"type":987,"prompt":988},"reflect-extend","reflection","Choose one unsolved problem from this layer. Explain it to a family member in two sentences, show them one example, and ask what they would guess. Why do you think such a simple-sounding question has not been solved?",{"id":990,"type":991,"title":992,"points":993},"cheat-extend","summary","Cheat sheet: primes in the wild",[994,995,996,997,998,999,1000,1001,1002],"**Cicadas:** 13- and 17-year cycles; co-prime cycles have the largest LCMs, so meetings with predators (and other broods) are rare.","**Cryptography:** multiplying huge primes is easy, factorising the product is very hard. RSA (1977) and other number-theory codes protect online payments.","**Mersenne primes:** 2ᵖ − 1 with p prime, but not always (2¹¹ − 1 = 23 × 89). Largest known prime (2024): 2¹³⁶²⁷⁹⁸⁴¹ − 1, 41 million digits.","**Perfect numbers:** 6, 28, 496, 8,128. Every even one is 2ᵖ⁻¹ × (2ᵖ − 1). No odd one is known.","**Goldbach:** every even number above 2 is a sum of two primes? Checked to 4 × 10¹⁸, unproved.","**Twin primes:** unproved; but infinitely many prime gaps are at most 246 (Zhang 2013, Maynard, Polymath).","**India:** Ramanujan (1,729, highly composite numbers); AKS test at IIT Kanpur (2002); Manjul Bhargava (Fields Medal 2014).","**Bases:** primality does not depend on the base; divisibility rules do.","**Puzzle tools:** remainders on division by 3, factor pairs, counting prime factors (zeros of 100!), odd factor counts for squares.",{"id":1004,"type":1005,"sourceIds":1006},"sources-extend","sources",[1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017],"prime-and-composite-wiki-periodical-cicada","prime-and-composite-gimps-mersenne","prime-and-composite-britannica-prime-number","prime-and-composite-wiki-twin-prime","prime-and-composite-khan-primes","prime-and-composite-mathisfun-prime","prime-and-composite-ncert-class6-prime-time","prime-and-composite-wiki-goldbach","prime-and-composite-wiki-perfect-number","prime-and-composite-wiki-rsa-numbers","prime-and-composite-wiki-aks",[1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017],"needs_review",{"generatedBy":1021,"notes":1022},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","168e40c83ab2f9c79b01efa491a40120e97580d4d0dc542310e3f3d090d2efb7",{"logic:practice":1025,"component:sort-game@1":1026,"component:factor-tree@1":1027,"component:match-pairs@1":1028,"component:prime-sieve@1":1029,"source:prime-and-composite-britannica-prime-number":1030,"source:prime-and-composite-gimps-mersenne":1031,"source:prime-and-composite-khan-primes":1032,"source:prime-and-composite-mathisfun-prime":1033,"source:prime-and-composite-ncert-class6-prime-time":1034,"source:prime-and-composite-wiki-aks":1035,"source:prime-and-composite-wiki-goldbach":1036,"source:prime-and-composite-wiki-perfect-number":1037,"source:prime-and-composite-wiki-periodical-cicada":1038,"source:prime-and-composite-wiki-rsa-numbers":1039,"source:prime-and-composite-wiki-twin-prime":1040},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","dfb598a254f20e70550d586121be496f7a546ba0254060fd97b95cb919dfd934","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","76baccc44f3b2ac333537045fea84801ea3c46f2d75f3db8a02355ef0f267041","8985127013a5a393bbde77d9c7d1e0757013c9dd047aa324fe51fa4706c2edfa","be41b20420652d001b4454ec3ac0d159a0139fb79e77363b0afc6f376ecc405c","7199c0a10c6364abf7af30d1869f37167826d837ce97a5b9cb0d8be7f2322212","ae2cb5b494e894b59fc2f34095d5ba7519d897f66195dc8bef83cbbcffa674b1","f6edb540745600a1873f0a5851335bf576bd243816f181e62085831fd55ac4ce","b817c8cf9495d0a940a4d3d0df7582557ce0c189585178e0abe4ad9236fd3c6d","7a1dff82432885b545643ddcf47d98dca9fcd489622acfd584a96c1aeb430121","4bcdd46e2d5e54a547257ece0de676d01594155b52169a2f7dae393a2f54d2f4","16058ea23b50d38df9f169941d978b866e6200e16f02efd314be2f5ffb56ea20","423e251e9045596d3b50b44811282dec93887e650518cf68ebb916aa479510f0","b0b179308ab5f683db390f67f15856137356f04cbd838b8129c6b89a3c2b8265",{"state":1042,"reviewer":1043,"selfReview":655,"reviewedAt":1044,"method":1045},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899596959]