[{"data":1,"prerenderedAt":1059},["ShallowReactive",2],{"layer:prime-and-composite:investigate":3},{"layer":4,"contentHash":1040,"dependencyHashes":1041,"approval":1053,"releaseId":1058},{"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":1035,"reviewStatus":1036,"authoring":1037},1,"prime-and-composite","en","investigate","Hunting patterns among the primes","Predict, test and decide: which prime patterns are real, and which ones fool you?","Test claims about primes like a mathematician: how fast primes thin out, the 6-column grid, last digits, twin prime hunts, why 3, 5, 7 stands alone, co-prime experiments, patterns that break, prime deserts and numbers with the most factors.",[13,14,15,16,17],"Make predictions about primes and test them with data and labs.","Explain why every prime above 3 is one more or one less than a multiple of 6, and why 3, 5, 7 is the only prime triplet.","Decide whether claims about primes, co-primes and divisibility are always, sometimes or never true, using counterexamples.","Build prime deserts of any length and describe how prime gaps and twin primes behave.","Discover the rule linking a prime factorisation to the number of factors.",50,{"title":20,"rows":21},"Lab plate",[22,25,28,31,34],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Time","≈ 50 minutes",{"label":29,"value":30},"Labs","6-column sieve, twin hunt, co-prime finder, sort, trees",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Key skill","One counterexample kills a claim",[38,42,48,54,72,112,137,142,169,173,187,192,195,208,224,246,258,263,268,281,313,318,323,326,340,364,368,381,386,389,404,408,413,418,421,432,462,475,486,491,494,549,619,623,636,641,644,673,684,689,692,713,760,770,774,779,782,792,802,823,829,834,839,882,1004,1008,1024],{"id":39,"type":40,"markdown":41},"intro-investigate","prose","This layer is a laboratory. Instead of being told facts about primes, you will **predict**, **test** and **decide** whether patterns are real. Mathematicians have done exactly this for over two thousand years, and some of the questions below are still being investigated today.\n\nThree rules for every investigation:\n\n1. **Predict first.** Write your guess before you test. A wrong prediction you can explain teaches more than a right guess you cannot.\n2. **Collect evidence.** Try many cases, including awkward ones: small numbers, big numbers, even and odd.\n3. **One counterexample kills a claim.** A pattern that works 40 times and fails once is **not** always true. But no number of examples can *prove* a pattern is always true; for that you need a reason, which is what the Deepen layer is for.",{"id":43,"type":44,"variant":45,"title":46,"markdown":47},"how-to-use-i","callout","observation","How to use this lesson","Keep a notebook open. Each chapter starts with a question and usually a prediction. Test it in the lab, then read what mathematicians have found. Mark every claim as **always**, **sometimes** or **never** true.",{"id":49,"type":50,"title":51,"eyebrow":52,"navLabel":53},"ch1","chapter","Do primes run out?","Chapter 01","1 Do primes run out?",{"id":55,"type":56,"prompt":57,"options":58,"explanation":71},"pred-thin","prediction","Between 1 and 100 there are 25 primes. **About how many primes do you expect between 1 and 1,000?**",[59,62,65,68],{"id":60,"label":61},"a","About 250: ten times as many",{"id":63,"label":64},"b","About 170",{"id":66,"label":67},"c","About 50",{"id":69,"label":70},"d","Exactly 25 more","**About 170.** In fact there are exactly **168** primes up to 1,000. Ten times the range gives fewer than seven times the primes. Primes **thin out** as numbers grow: 25.0% of the numbers up to 100 are prime, but only 16.8% of those up to 1,000.\n\nWhy? A big number has many more possible factors to \"catch\" it. A number near 1,000 must avoid being a multiple of every prime up to 31 to be prime; a number near 100 only has to dodge 2, 3, 5 and 7.",{"id":73,"type":74,"caption":75,"columns":76,"rows":81},"table-pi","table","How many primes up to 10, 100, 1,000, …? (Counted by computer.)",[77,78,79,80],"Up to","Number of primes","Share that are prime","Average gap between primes",[82,87,92,97,102,107],[83,84,85,86],"10","4","40.0%","about 2.5",[88,89,90,91],"100","25","25.0%","about 4.0",[93,94,95,96],"1,000","168","16.8%","about 6.0",[98,99,100,101],"10,000","1,229","12.3%","about 8.1",[103,104,105,106],"100,000","9,592","9.6%","about 10.4",[108,109,110,111],"1,000,000","78,498","7.8%","about 12.7",{"id":113,"type":114,"title":115,"note":116,"scale":117,"rungs":118},"ladder-pi","ladder","The prime count grows, but more and more slowly","Log scale: each step up is ten times bigger. The number of primes keeps growing, but each tenfold jump in range adds fewer than tenfold primes.","log",[119,122,125,128,131,134],{"label":120,"value":121,"display":84},"Primes up to 10",4,{"label":123,"value":124,"display":89},"Primes up to 100",25,{"label":126,"value":127,"display":94},"Primes up to 1,000",168,{"label":129,"value":130,"display":99},"Primes up to 10,000",1229,{"label":132,"value":133,"display":104},"Primes up to 100,000",9592,{"label":135,"value":136,"display":109},"Primes up to 1,000,000",78498,{"id":138,"type":44,"variant":139,"title":140,"markdown":141},"aha-never-run-out","aha","Thinning out is not running out","The table shows primes becoming rarer, and the gaps between them growing on average. It is natural to wonder whether they eventually stop. They never do: there are **infinitely many** primes. Euclid proved it around 300 BCE with an argument you will meet in the Deepen layer. Rarer and rarer, but never finished.",{"id":143,"type":74,"caption":144,"columns":145,"rows":148},"table-hundreds","Primes in each hundred up to 1,000",[146,147,146,147],"Block","Primes",[149,153,158,161,165],[150,89,151,152],"1–100","501–600","14",[154,155,156,157],"101–200","21","601–700","16",[159,157,160,152],"201–300","701–800",[162,157,163,164],"301–400","801–900","15",[166,167,168,152],"401–500","17","901–1000",{"id":170,"type":44,"variant":45,"title":171,"markdown":172},"obs-bumpy","Thinning, but bumpy","The counts do not fall smoothly: 16 primes in 201–300 and 17 in 401–500, then 14, 16, 14, 15, 14. Primes follow an overall trend but jump around locally. This mix of order and surprise is a big part of why mathematicians find them fascinating.",{"id":174,"type":175,"itemId":176,"prompt":177,"check":178,"hints":182,"feedback":184},"prac-101-200","practice","prime-and-composite.investigate-primes-101-200","Use the table: how many primes are there **from 101 to 200**?",{"kind":179,"answer":180,"tolerance":181},"number",21,0,[183],"Look up the row 101–200.",{"correct":185,"incorrect":186},"Yes: 21 primes, compared with 25 in the first hundred.","The table shows 21 primes from 101 to 200. The first hundred has 25.",{"id":188,"type":50,"title":189,"eyebrow":190,"navLabel":191},"ch2","The 6-column grid","Chapter 02","2 The 6-column grid",{"id":193,"type":40,"markdown":194},"six-col","In Discover and Understand you drew the numbers in rows of 10. Now rearrange them in **rows of 6**:\n\n1, 2, 3, 4, 5, 6\n7, 8, 9, 10, 11, 12\n13, 14, 15, 16, 17, 18\n…\n\nBefore you look, predict: where will the primes land?",{"id":196,"type":56,"prompt":197,"options":198,"explanation":207},"pred-six","On a grid with **6 columns**, which columns will contain primes after the first row?",[199,201,203,205],{"id":60,"label":200},"All six columns, scattered randomly",{"id":63,"label":202},"Only the columns under 1 and 5",{"id":66,"label":204},"Only the odd columns: under 1, 3 and 5",{"id":69,"label":206},"Only the column under 1","**Only the columns under 1 and 5.** Every number in the column under 2, 4 or 6 is even. Every number in the column under 3 is a multiple of 3. So after 2 and 3 themselves, primes can only live in the columns of numbers that are **one more** or **one less** than a multiple of 6.\n\nBut careful: not every number in those two columns is prime. Up to 100 the composites hiding there are 25, 35, 49, 55, 65, 77, 85, 91, 95. Every one of them is built from primes of 5 or more (25 = 5 × 5, 35 = 5 × 7, 91 = 7 × 13).",{"id":209,"type":210,"component":211,"componentVersion":5,"config":212,"objective":218,"textAlternative":219,"help":220},"lab-six-col","interactive","prime-sieve",{"max":213,"columns":214,"modes":215,"rounds":214},100,6,[216,217],"sieve","hunt","Sieve the numbers 1 to 100 on a 6-column grid and see which columns the primes fall into.","The lab lays out 1 to 100 in rows of 6. After sieving:\n\n- Column under **1** (1, 7, 13, 19, 25, …, 97): holds the primes 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97 and the composites 25, 49, 55, 85, 91.\n- Column under **5** (5, 11, 17, 23, 29, …, 95): holds 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89 and the composites 35, 65, 77, 95.\n- Columns under **2, 4, 6** contain only even numbers; the only prime among them is 2.\n- Column under **3** contains only multiples of 3; the only prime is 3.\n\nSo every prime bigger than 3 is **one less or one more than a multiple of 6** (6k − 1 or 6k + 1). The primes look like two straight stripes. In hunt mode you can use this: skip four of the six columns.",{"hints":221},[222,223],"Colour the columns under 2, 4 and 6 first. What do they have in common?","Is 25 in a prime column? Is it prime?",{"id":225,"type":175,"itemId":226,"prompt":227,"check":228,"hints":240,"feedback":243},"prac-six","prime-and-composite.investigate-6k-claim","Which statement is **always** true?",{"kind":229,"options":230,"correct":239},"choice",[231,233,235,237],{"id":60,"label":232},"Every number of the form 6k + 1 is prime.",{"id":63,"label":234},"Every prime bigger than 3 is of the form 6k − 1 or 6k + 1.",{"id":66,"label":236},"Every number of the form 6k − 1 is prime.",{"id":69,"label":238},"Every odd number is of the form 6k + 1.",[63],[241,242],"Test (a) with k = 4.","The other columns are multiples of 2 or 3.",{"correct":244,"incorrect":245},"Yes. The other four columns are multiples of 2 or 3, so primes above 3 can only be in the 6k ± 1 columns. The reverse is false: 25 = 6 × 4 + 1 is composite.","Only (b) is always true. 25 = 6 × 4 + 1 and 35 = 6 × 6 − 1 are composite, so (a) and (c) fail; 9 is odd but is 6 + 3.",{"id":247,"type":56,"prompt":248,"options":249,"explanation":257},"pred-four","Now use a grid with **4 columns**. Apart from 2, which columns can hold primes, and will the primes be shared **evenly** between them?",[250,252,253,255],{"id":60,"label":251},"Columns under 1 and 3, shared about evenly",{"id":63,"label":206},{"id":66,"label":254},"Columns under 1 and 3, but almost all in the column under 1",{"id":69,"label":256},"All four columns","**Columns under 1 and 3, shared about evenly.** The columns under 2 and 4 are all even. Every odd prime is either **4k + 1** (5, 13, 17, 29, …) or **4k + 3** (3, 7, 11, 19, …). Up to 100 the split is 11 to 13; up to 1,000 it is 80 to 87.\n\nLook closely and the 4k + 3 column is slightly ahead most of the time. This small lead is called **Chebyshev’s bias**, after the Russian mathematician who noticed it in 1853. The two columns swap the lead now and then, first at 26,861, so the bias is real but not permanent.",{"id":259,"type":44,"variant":260,"title":261,"markdown":262},"ex-fermat","example","A secret of the 4k + 1 primes","Primes in the 4k + 1 column can always be written as the sum of two square numbers: 5 = 1 + 4, 13 = 4 + 9, 17 = 1 + 16, 29 = 4 + 25, 37 = 1 + 36. Primes in the 4k + 3 column never can. Pierre de Fermat stated this in 1640; Euler published the first proof about a hundred years later. Try 41, 53 and 61 yourself.",{"id":264,"type":50,"title":265,"eyebrow":266,"navLabel":267},"ch3","What digit do primes end in?","Chapter 03","3 Last digits",{"id":269,"type":56,"prompt":270,"options":271,"explanation":280},"pred-lastdigit","Look at the primes up to 1,000, apart from 2 and 5. **Which last digits can they have?**",[272,274,276,278],{"id":60,"label":273},"Any digit",{"id":63,"label":275},"Only 1, 3, 7 or 9",{"id":66,"label":277},"Only 1 or 7",{"id":69,"label":279},"Only odd digits: 1, 3, 5, 7, 9","**Only 1, 3, 7 or 9.** A number ending in 0, 2, 4, 6 or 8 is even, and one ending in 0 or 5 is a multiple of 5. So only 2 and 5 themselves escape.\n\nOf the 168 primes up to 1,000, **40** end in 1, **42** end in 3, **46** end in 7 and **38** end in 9, plus 2 and 5. The four endings share the primes roughly equally. Mathematicians have proved that in the long run each ending gets a quarter of the primes, although in 2016 two researchers noticed a surprising bias: a prime ending in 1 is less often followed by another prime ending in 1 than you would expect by chance.",{"id":282,"type":74,"caption":283,"columns":284,"rows":288},"table-lastdig","Last digits of the primes up to 1,000",[285,286,287],"Last digit","How many primes","Examples",[289,293,297,301,305,309],[290,291,292],"1","40","11, 31, 41, 61, 71, 101",[294,295,296],"3","42","3, 13, 23, 43, 53, 73",[298,299,300],"7","46","7, 17, 37, 47, 67, 97",[302,303,304],"9","38","19, 29, 59, 79, 89, 109",[306,307,308],"2 or 5","2","Only 2 and 5 themselves",[310,311,312],"0, 4, 6 or 8","0","None: these are all even",{"id":314,"type":44,"variant":315,"title":316,"markdown":317},"careful-ending","careful","The right ending is not enough","Ending in 1, 3, 7 or 9 is **necessary** for a prime above 5, but not **sufficient**. 21, 27, 33, 39, 49, 51, 57, 63, 69, 77, 81, 87, 91, 93 and 99 all have \"prime-looking\" endings. The last digit can rule a number out, never in.",{"id":319,"type":50,"title":320,"eyebrow":321,"navLabel":322},"ch4","Hunting twin primes","Chapter 04","4 Twin prime hunt",{"id":324,"type":40,"markdown":325},"twins-i","Twin primes are prime pairs that differ by 2, such as 41 and 43. Use the lab to hunt them up to 200, and keep a tally of how many you find in each hundred. Then compare with the data below, which a computer counted up to 1,000.",{"id":327,"type":210,"component":211,"componentVersion":5,"config":328,"objective":334,"textAlternative":335,"help":336},"lab-twins",{"max":329,"columns":330,"modes":331,"rounds":333},200,10,[332],"twins",8,"Find all the twin prime pairs up to 200 and notice where they sit on the grid.","In **twins** mode the lab shows 1 to 200 in rows of 10 and asks you to tap both members of each twin prime pair.\n\nUp to 200 there are **15 pairs**: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199).\n\nPatterns to notice:\n- After (3, 5), every pair straddles a **multiple of 6**: 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198.\n- On a 10-column grid, twins after (5, 7) always end in 1 and 3, 7 and 9, or 9 and 1.\n- There are long stretches with none, such as from 150 to 178.",{"hints":337},[338,339],"Look either side of each multiple of 6.","Both numbers must be prime: 25 spoils (23, 25).",{"id":341,"type":74,"caption":342,"columns":343,"rows":346},"table-twins-hundreds","Twin prime pairs by the hundred in which the smaller prime lies, up to 1,000",[344,345,344,345],"Hundred","Twin pairs",[347,351,354,357,361],[348,349,350,294],"0–99","8","500–599",[352,298,353,294],"100–199","600–699",[355,84,356,311],"200–299","700–799",[358,307,359,360],"300–399","800–899","5",[362,294,363,311],"400–499","900–999",{"id":365,"type":44,"variant":45,"title":366,"markdown":367},"obs-twins","Twins thin out even faster","Up to 1,000 there are 35 twin pairs, with none at all in 700–799 or 900–999. Twins get rarer faster than primes do. Yet computers have found twin primes with hundreds of thousands of digits. Whether they go on forever is the **twin prime conjecture**, one of the oldest unsolved problems in mathematics.",{"id":369,"type":56,"prompt":370,"options":371,"explanation":380},"pred-middle","The number **between** a pair of twin primes (like 12 between 11 and 13, or 30 between 29 and 31). Apart from the pair (3, 5), what is always true about it?",[372,374,376,378],{"id":60,"label":373},"It is a multiple of 4",{"id":63,"label":375},"It is a multiple of 6",{"id":66,"label":377},"It is a multiple of 10",{"id":69,"label":379},"Nothing: it varies","**It is a multiple of 6.** The middle number sits between two odd numbers, so it is **even**. Also, of any three numbers in a row, one is a multiple of 3. The two twins are prime and bigger than 3, so neither of them is that multiple of 3: it must be the middle number. Even and a multiple of 3 means a multiple of 6. This is exactly the 6-column pattern from Chapter 2.",{"id":382,"type":50,"title":383,"eyebrow":384,"navLabel":385},"ch5","Why 3, 5, 7 stands alone","Chapter 05","5 Prime triplets",{"id":387,"type":40,"markdown":388},"triplet-i","3, 5 and 7 are three primes in a row, each 2 apart. Can you find another such **prime triplet** of the form n, n + 2, n + 4?\n\nTry some: 5, 7, 9 (9 = 3 × 3). 11, 13, 15 (15 = 3 × 5). 17, 19, 21 (21 = 3 × 7). 29, 31, 33 (33 = 3 × 11). 41, 43, 45. Each time one of the three is a multiple of 3. Is that a coincidence?",{"id":390,"type":74,"caption":391,"columns":392,"rows":397},"table-triplet","Remainders when n, n + 2 and n + 4 are divided by 3",[393,394,395,396],"If n leaves remainder…","n + 2 leaves","n + 4 leaves","Which one is a multiple of 3?",[398,400,402],[311,307,290,399],"n itself",[290,311,307,401],"n + 2",[307,290,311,403],"n + 4",{"id":405,"type":44,"variant":139,"title":406,"markdown":407},"aha-triplet","Never a coincidence","The table covers **every** possible n, because any whole number leaves remainder 0, 1 or 2 when divided by 3. In every row, one of n, n + 2 and n + 4 is a multiple of 3. For that number to be prime as well, it must be 3 itself. So the only possible triplet is **3, 5, 7**.\n\nThis is your first taste of a **proof**: instead of checking examples forever, you checked every *type* of number at once.",{"id":409,"type":44,"variant":410,"title":411,"markdown":412},"nuance-triplet","nuance","Other meanings of \"prime triplet\"","Some books use \"prime triplet\" for three primes packed as tightly as possible, which after 3, 5, 7 means patterns like (p, p + 2, p + 6) or (p, p + 4, p + 6): for example 5, 7, 11 and 7, 11, 13. Those do keep appearing. In this topic, \"prime triplet\" means three primes each 2 apart, and there is only one.",{"id":414,"type":50,"title":415,"eyebrow":416,"navLabel":417},"ch6","Co-prime experiments","Chapter 06","6 Co-prime experiments",{"id":419,"type":40,"markdown":420},"coprime-exp","Two numbers are co-prime when their highest common factor is 1. Test these claims yourself before reading the verdicts in the table. For each one, try at least five examples, including small ones.\n\n1. Two consecutive numbers are always co-prime.\n2. Two consecutive **odd** numbers are always co-prime.\n3. Two consecutive **even** numbers are always co-prime.\n4. If a and b are co-prime, then a and a + b are co-prime.\n5. A prime and any other number are co-prime.\n6. Two composite numbers are never co-prime.",{"id":422,"type":210,"component":211,"componentVersion":5,"config":423,"objective":426,"textAlternative":427,"help":428},"lab-coprime",{"max":213,"columns":330,"modes":424,"rounds":333},[425],"coprime","Choose a number and light up every number up to 100 that is co-prime with it, then spot the pattern.","In **co-prime** mode the lab picks a target number and asks you to tap every number on the 1 to 100 grid that is co-prime with it (shares no factor except 1).\n\n- Target **10** = 2 × 5: the co-prime numbers are exactly those ending in 1, 3, 7 or 9. There are 40 of them up to 100.\n- Target **12** = 2² × 3: the co-prime numbers are those that are neither even nor a multiple of 3, which fall in two columns on a 6-column grid. There are 33 up to 100.\n- Target **7** (prime): every number except the multiples of 7 is co-prime with it: 100 − 14 = 86 numbers.\n- Target **30** = 2 × 3 × 5: only 26 numbers up to 100 are co-prime with it, including 1, 7, 11, 13, 17, 19, 23, 29, 31 and 49.\n\nThe more different primes a target has, the fewer numbers are co-prime with it. The number's neighbours, one below and one above, are always co-prime with it.",{"hints":429},[430,431],"Split the target into primes. Anything sharing one of those primes is out.","Start by tapping the numbers next to the target.",{"id":433,"type":74,"caption":434,"columns":435,"rows":439},"table-coprime-claims","Verdicts on the six co-prime claims",[436,437,438],"Claim","Verdict","Evidence or reason",[440,444,447,451,454,458],[441,442,443],"1. Consecutive numbers","Always","A common factor would divide their difference, which is 1. So the HCF is 1.",[445,442,446],"2. Consecutive odd numbers","A common factor would divide the difference 2, so it is 1 or 2. Both numbers are odd, so it is not 2.",[448,449,450],"3. Consecutive even numbers","Never","Both are even, so they share 2. Example: 10 and 12.",[452,442,453],"4. a, b co-prime ⇒ a and a + b co-prime","A factor of a and a + b also divides (a + b) − a = b. So it is a common factor of a and b: only 1.",[455,456,457],"5. A prime and any number","Sometimes","7 and 30 are co-prime, but 7 and 35 are not. It fails when the number is a multiple of the prime.",[459,460,461],"6. Two composites are never co-prime","False (sometimes co-prime)","8 and 15, 4 and 9 and 25 and 36 are co-prime pairs of composites.",{"id":463,"type":56,"prompt":464,"options":465,"explanation":474},"pred-coprime-chance","Pick two different numbers from 1 to 10. There are 45 possible pairs. **About how many of them are co-prime?**",[466,468,470,472],{"id":60,"label":467},"About 10",{"id":63,"label":469},"About 20",{"id":66,"label":471},"About 30",{"id":69,"label":473},"All 45","**About 30.** Exactly **31** of the 45 pairs are co-prime, which is about 69%. Most pairs of small numbers share no factor! As the range grows, the share of co-prime pairs settles near 61%. The exact limit is 6 ÷ π², about 0.608. Surprisingly, it involves π, the number from circles, through a famous discovery by the Swiss mathematician Leonhard Euler.",{"id":476,"type":175,"itemId":477,"prompt":478,"check":479,"hints":480,"feedback":483},"prac-coprime-12","prime-and-composite.investigate-coprime-to-12","How many numbers from **1 to 12** are co-prime with **12**?",{"kind":179,"answer":121,"tolerance":181},[481,482],"12 = 2 × 2 × 3. Cross out every multiple of 2 or 3.","Do not forget 1.",{"correct":484,"incorrect":485},"Yes: 1, 5, 7 and 11. Four numbers. They are exactly the 6k ± 1 numbers again.","Only 1, 5, 7 and 11 share no factor with 12. Every other number up to 12 is even or a multiple of 3.",{"id":487,"type":50,"title":488,"eyebrow":489,"navLabel":490},"ch7","Is it always true? Patterns that fool you","Chapter 07","7 Pattern traps",{"id":492,"type":40,"markdown":493},"traps-intro","Primes are famous for luring people into false patterns. Here are four real ones. For each, test a few cases and decide whether you believe it before reading on.",{"id":495,"type":496,"title":497,"prompt":498,"options":499},"explorer-traps","explorer","Four patterns that look perfect","Choose a pattern to see how long it lasts and where it breaks.",[500,513,526,538],{"id":501,"label":502,"chain":503,"badge":509,"note":512},"euler","n × n + n + 41",[504,505,506,507,508],"n = 0 → 41","n = 1 → 43","n = 2 → 47","… all prime …","n = 40 → 1,681",{"text":510,"tone":511},"Breaks at n = 40","no","Leonhard Euler noticed in 1772 that n × n + n + 41 gives a prime for n = 0, 1, 2, … all the way to 39: forty primes in a row. At n = 40 it gives 1,681 = 41 × 41. It must fail there, because every term is 40 × 40 + 40 + 41 = 40 × 41 + 41 = 41 × 41. Forty successes, then one failure, and the claim \"always prime\" is dead.",{"id":514,"label":515,"chain":516,"badge":523,"note":525},"threes","31, 331, 3331, …",[517,518,519,520,521,522],"31","331","3,331","33,331","… 33,333,331","333,333,331",{"text":524,"tone":511},"Breaks at the 8th term","31, 331, 3,331, 33,331, 333,331, 3,333,331 and 33,333,331 are all prime: seven in a row. The eighth, 333,333,331, equals 17 × 19,607,843. You would never find that by hand, which is exactly why patterns need proof, not just examples.",{"id":527,"label":528,"chain":529,"badge":535,"note":537},"mersenne","2 to a prime power, − 1",[530,531,532,533,534],"2² − 1 = 3","2³ − 1 = 7","2⁵ − 1 = 31","2⁷ − 1 = 127","2¹¹ − 1 = 2,047",{"text":536,"tone":511},"Breaks at 11","Take a prime p and work out 2 multiplied by itself p times, minus 1. For p = 2, 3, 5 and 7 you get the primes 3, 7, 31 and 127. It is tempting to think this always gives a prime, but 2¹¹ − 1 = 2,047 = 23 × 89. The primes that *do* appear this way, called Mersenne primes, are among the largest primes known (see Extend).",{"id":539,"label":540,"chain":541,"badge":546,"note":548},"repunit","11, 111, 1111, …",[542,543,544,545],"11 prime","111 = 3 × 37","1,111 = 11 × 101","11,111 = 41 × 271",{"text":547,"tone":511},"Breaks at once","11 is prime, so perhaps all numbers made only of 1s are prime? 111 has digit sum 3, so it is divisible by 3: 111 = 3 × 37. A number of 1s can only be prime if its number of digits is itself prime, but even that is not enough: 11,111 has 5 digits and equals 41 × 271. Numbers of all 1s are called repunits.",{"id":550,"type":210,"component":551,"componentVersion":5,"config":552,"objective":613,"textAlternative":614,"help":615},"lab-always-sort","sort-game",{"prompt":553,"bins":554,"items":564,"seconds":181},"Is each statement always, sometimes or never true? Test it with examples.",[555,558,561],{"id":556,"label":557},"always","Always true",{"id":559,"label":560},"sometimes","Sometimes true",{"id":562,"label":563},"never","Never true",[565,569,573,577,581,585,589,593,597,601,605,609],{"id":566,"label":567,"bin":559,"why":568},"s1","The sum of two primes is even.","3 + 5 = 8 is even, but 2 + 3 = 5 is odd. It fails whenever one of the primes is 2.",{"id":570,"label":571,"bin":556,"why":572},"s2","The product of two primes is composite.","p × q has at least the factors 1, p, q and p × q (or 1, p, p × p when p = q).",{"id":574,"label":575,"bin":556,"why":576},"s3","A prime bigger than 2 is odd.","Any even number bigger than 2 has 2 as an extra factor.",{"id":578,"label":579,"bin":562,"why":580},"s4","The sum of three consecutive numbers is prime.","n + (n + 1) + (n + 2) = 3 × (n + 1), a multiple of 3 bigger than 3.",{"id":582,"label":583,"bin":559,"why":584},"s5","A number ending in 7 is prime.","17 and 37 are prime, but 27 and 57 are not.",{"id":586,"label":587,"bin":556,"why":588},"s6","Two consecutive numbers are co-prime.","Any common factor would divide their difference, 1.",{"id":590,"label":591,"bin":562,"why":592},"s7","A square number (4, 9, 16, …) is prime.","n × n has factors 1, n and n × n: at least three.",{"id":594,"label":595,"bin":559,"why":596},"s8","Adding 1 to a prime gives a composite.","3 + 1 = 4 is composite, but 2 + 1 = 3 is prime.",{"id":598,"label":599,"bin":559,"why":600},"s9","A number divisible by 4 and by 6 is divisible by 24.","24 and 48 work, but 12 is divisible by 4 and 6 and not by 24.",{"id":602,"label":603,"bin":556,"why":604},"s10","A number divisible by 3 and by 5 is divisible by 15.","3 and 5 are co-prime, so both tests together mean divisible by 15.",{"id":606,"label":607,"bin":562,"why":608},"s11","Two even numbers are co-prime.","They share the factor 2.",{"id":610,"label":611,"bin":556,"why":612},"s12","A number with exactly three factors is a square.","Only squares have an odd number of factors; three factors means the square of a prime, like 4, 9, 25, 49.","Decide whether each claim about primes and factors is always, sometimes or never true, using examples and counterexamples.","This game has 12 statements and three bins.\n\n**Always true:** the product of two primes is composite; a prime bigger than 2 is odd; two consecutive numbers are co-prime; a number divisible by 3 and 5 is divisible by 15; a number with exactly three factors is a square.\n\n**Sometimes true:** the sum of two primes is even (fails with 2 + 3); a number ending in 7 is prime (27 is not); adding 1 to a prime gives a composite (2 + 1 = 3 is prime); divisible by 4 and 6 means divisible by 24 (12 is a counterexample).\n\n**Never true:** the sum of three consecutive numbers is prime (it is always 3 times the middle number); a square is prime; two even numbers are co-prime.\n\nTo show \"sometimes\", give one example and one counterexample. To show \"always\" or \"never\", you need a reason that covers every case.",{"hints":616},[617,618],"Always test the number 2: it breaks many prime patterns.","Write three consecutive numbers as n, n + 1, n + 2 and add them.",{"id":620,"type":44,"variant":315,"title":621,"markdown":622},"careful-examples","Examples are not proof","Euler’s formula worked 40 times in a row and still failed. When you think you have found an \"always\" pattern, the next step is to look for a **reason** that covers every number, as we did for the prime triplet. Until then, call it a **conjecture**: a guess supported by evidence.",{"id":624,"type":56,"prompt":625,"options":626,"explanation":635},"pred-odd-sum","Every even number from 4 to 100 can be written as the sum of two primes (try a few!). What about **odd** numbers? Which odd numbers above 3 are the sum of two primes?",[627,629,631,633],{"id":60,"label":628},"All of them",{"id":63,"label":630},"Only those that are 2 more than a prime",{"id":66,"label":632},"None of them",{"id":69,"label":634},"Only the odd primes","**Only those that are 2 more than a prime.** Odd = odd + even, and the only even prime is 2. So an odd number n is a sum of two primes exactly when **n − 2 is prime**: 9 = 2 + 7 works, but 11 − 2 = 9 is not prime, so 11 cannot be done. Below 40 the odd numbers above 3 that fail are 11, 17, 23, 27, 29, 35 and 37.\n\nThe even case is far harder. That every even number above 2 is a sum of two primes is **Goldbach's conjecture**, checked by computer to enormous sizes but never proved. You will meet it in the Extend layer.",{"id":637,"type":50,"title":638,"eyebrow":639,"navLabel":640},"ch8","Prime deserts and prime gaps","Chapter 08","8 Prime gaps",{"id":642,"type":40,"markdown":643},"gaps-prose","The **gap** between two neighbouring primes is their difference. Up to 100, the gaps are small: mostly 2, 4 or 6. The biggest is between **89 and 97**, a gap of 8, with seven composites (90 to 96) in a row.\n\nUp to 1,000 the biggest gap is **20**, between 887 and 907. Can gaps be as big as you like? Here is a clever way to build a \"prime desert\" of any length.\n\nMultiply 2 × 3 × 4 × 5 × 6 = 720. Now look at 722, 723, 724, 725, 726:\n\n- 722 = 720 + 2 is divisible by 2 (both parts are).\n- 723 = 720 + 3 is divisible by 3.\n- 724 = 720 + 4 is divisible by 4.\n- 725 = 720 + 5 is divisible by 5.\n- 726 = 720 + 6 is divisible by 6.\n\nFive composites in a row, guaranteed, without testing any of them. Multiply up to 11 instead and you get ten composites in a row. There are prime deserts as long as you want.",{"id":645,"type":74,"caption":646,"columns":647,"rows":651},"table-gaps","Record prime gaps below 1,000: each gap is bigger than any before it",[648,649,650],"Gap","Between","Composites in a row",[652,654,656,658,661,663,666,669],[290,653,311],"2 and 3",[307,655,290],"3 and 5",[84,657,294],"7 and 11",[659,660,360],"6","23 and 29",[349,662,298],"89 and 97",[152,664,665],"113 and 127","13",[667,668,167],"18","523 and 541",[670,671,672],"20","887 and 907","19",{"id":674,"type":175,"itemId":675,"prompt":676,"check":677,"hints":678,"feedback":681},"prac-gap","prime-and-composite.investigate-desert-length","Using the same trick, 2 × 3 × 4 × 5 × 6 × 7 = 5,040. **How many composite numbers in a row** does the trick guarantee, starting from 5,042?",{"kind":179,"answer":214,"tolerance":181},[679,680],"5,040 + 2 is divisible by 2, 5,040 + 3 by 3, …","Where does the trick stop working?",{"correct":682,"incorrect":683},"Yes: 5,042 to 5,047 are divisible by 2, 3, 4, 5, 6 and 7 in turn: six composites in a row.","The trick gives 5,040 + 2, + 3, + 4, + 5, + 6, + 7: six numbers, each divisible by the number added. So six composites in a row.",{"id":685,"type":50,"title":686,"eyebrow":687,"navLabel":688},"ch9","Which numbers have the most factors?","Chapter 09","9 Counting factors",{"id":690,"type":40,"markdown":691},"count-prose","Primes have the fewest factors possible (two). At the other extreme, some numbers are crammed with factors. Investigate:\n\n- Which numbers have **exactly 3** factors? Try 4, 9, 25 and 49. Can you find any others under 100?\n- Which numbers have an **odd** number of factors?\n- Can you predict the number of factors from the prime factorisation?\n\nUse the factor tree lab to factorise each number, then count its factors and look for a rule.",{"id":693,"type":210,"component":694,"componentVersion":5,"config":695,"objective":707,"textAlternative":708,"help":709},"lab-tree-count","factor-tree",{"numbers":696,"showIndexForm":706},[697,698,699,700,701,213,702,703,704,705],16,24,36,48,72,120,144,180,360,true,"Factorise numbers with many factors, then compare the index form with the number of factors to find a rule.","The lab builds a factor tree for each number and writes the result in index form. Record the number of factors next to each one.\n\n- 16 = 2⁴: 5 factors\n- 24 = 2³ × 3: 8 factors\n- 36 = 2² × 3²: 9 factors\n- 48 = 2⁴ × 3: 10 factors\n- 72 = 2³ × 3²: 12 factors\n- 100 = 2² × 5²: 9 factors\n- 120 = 2³ × 3 × 5: 16 factors\n- 144 = 2⁴ × 3²: 15 factors\n- 180 = 2² × 3² × 5: 18 factors\n- 360 = 2³ × 3² × 5: 24 factors\n\nLook at 16 = 2⁴ with 5 factors, 36 = 2² × 3² with 9 = 3 × 3 factors, and 72 = 2³ × 3² with 12 = 4 × 3 factors. The pattern: add 1 to each power and multiply. 360 = 2³ × 3² × 5 gives 4 × 3 × 2 = 24 factors. The Deepen layer explains why.",{"hints":710},[711,712],"Count the factors by listing factor pairs.","Compare 16 = 2⁴ (5 factors) with 81 = 3⁴. How many factors does 81 have?",{"id":714,"type":74,"caption":715,"columns":716,"rows":721},"table-counts","Number of factors and prime factorisation: spot the rule",[717,718,719,720],"Number","Index form","Powers + 1","Number of factors",[722,724,728,730,733,737,741,745,749,753,755],[349,723,84,84],"2³",[725,726,727,659],"12","2² × 3","3 × 2",[157,729,360,360],"2⁴",[667,731,732,659],"2 × 3²","2 × 3",[734,735,736,349],"30","2 × 3 × 5","2 × 2 × 2",[738,739,740,302],"36","2² × 3²","3 × 3",[742,743,744,83],"48","2⁴ × 3","5 × 2",[746,747,748,725],"60","2² × 3 × 5","3 × 2 × 2",[750,751,752,725],"72","2³ × 3²","4 × 3",[88,754,740,302],"2² × 5²",[756,757,758,759],"360","2³ × 3² × 5","4 × 3 × 2","24",{"id":761,"type":56,"prompt":762,"options":763,"explanation":769},"pred-three-factors","How many numbers **below 100** have exactly **three** factors?",[764,766,767,768],{"id":60,"label":765},"None",{"id":63,"label":84},{"id":66,"label":302},{"id":69,"label":89},"**4:** they are 4, 9, 25, 49, the squares of the primes 2, 3, 5 and 7. A number with three factors has an odd number of factors, so it is a square, n × n. Its factors include 1, n and n × n; to have no more, n must be prime. The next one is 11 × 11 = 121.",{"id":771,"type":44,"variant":45,"title":772,"markdown":773},"obs-900","Champions of factors","Below 1,000 the number with the most factors is **840 = 2³ × 3 × 5 × 7**, with 32 factors. The Indian mathematician **Srinivasa Ramanujan** studied such numbers, which he called **highly composite numbers**, in a long paper published in 1915. They are the opposite of primes: numbers with more factors than any smaller number, like 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720 and 840.",{"id":775,"type":50,"title":776,"eyebrow":777,"navLabel":778},"ch10","Testing divisibility tricks","Chapter 10","10 Testing the rules",{"id":780,"type":40,"markdown":781},"div-inv","Divisibility rules are claims too, so test them like any other.\n\n- **Does the digit-sum trick work for 7?** 16 has digit sum 7 but 16 is not divisible by 7; 21 is divisible by 7 but its digit sum is 3. So no: the digit-sum trick is special to 3 and 9.\n- **Does the \"last two digits\" trick work for 3?** 115 ends in 15, which is divisible by 3, but 115 is not. So no: that trick is special to 4 (and 25).\n- **Reverse the digits.** 82 − 28 = 54; 731 − 137 = 594; 5,020 − 205 = 4,815. Divide each by 9. Is a number minus its reverse always a multiple of 9? Test more cases, then look for the reason in Deepen.",{"id":783,"type":784,"title":785,"problem":786,"steps":787},"we-reverse","worked_example","A number minus its reverse","Test the claim \"a number minus its reverse is always divisible by 9\" on 4,213.",[788,789,790,791],"The reverse of 4,213 is 3,124.","4,213 − 3,124 = 1,089.","Digit sum of 1,089 is 1 + 0 + 8 + 9 = 18, a multiple of 9. So 1,089 is divisible by 9: 1,089 = 9 × 121.","Why it works: a number and its reverse have the **same digits**, so the same digit sum. Each leaves the same remainder when divided by 9, so their difference leaves remainder 0.",{"id":793,"type":784,"title":794,"problem":795,"steps":796},"we-abcabc","The 1,001 trick","Pick any three-digit number, say 358, and write it twice: 358,358. Show that the result is always divisible by 7, 11 and 13.",[797,798,799,800,801],"358,358 = 358 × 1,000 + 358 = 358 × 1,001.","Factorise 1,001 by trial division: 1,001 ÷ 7 = 143, and 143 = 11 × 13. So 1,001 = 7 × 11 × 13.","So 358,358 = 358 × 7 × 11 × 13. It is divisible by 7, by 11 and by 13, and also by 358.","The same works for **any** three-digit number abc, because abcabc = abc × 1,001. This is a pattern with a reason, so it is always true.","Try it as a magic trick: ask a friend to write any \"double\" number and divide it by 7, then 11, then 13. They will end with their original three-digit number, with no remainders along the way.",{"id":803,"type":175,"itemId":804,"prompt":805,"check":806,"hints":817,"feedback":820},"prac-div-inv","prime-and-composite.investigate-rule-for-12","Which test correctly checks whether a number is divisible by **12**?",{"kind":229,"options":807,"correct":816},[808,810,812,814],{"id":60,"label":809},"Divisible by 2 and by 6",{"id":63,"label":811},"Divisible by 3 and by 4",{"id":66,"label":813},"Last two digits divisible by 12",{"id":69,"label":815},"Digit sum divisible by 12",[63],[818,819],"12 = 3 × 4. Are 3 and 4 co-prime?","Test (a) on 18.",{"correct":821,"incorrect":822},"Right. 3 and 4 are co-prime and 3 × 4 = 12, so passing both tests means divisible by 12.","Use 3 and 4, which are co-prime. (a) fails on 18 (divisible by 2 and 6 but not 12), (c) fails on 112 (ends in 12, but 112 is not divisible by 12) and (d) fails on 39 (digit sum 12, but 39 is not divisible by 12).",{"id":824,"type":825,"conceptId":826,"relation":827,"explanation":828},"conn-patterns","connection","patterns","contrasts_with","Unlike squares or multiples, primes resist every simple rule: the traps in this layer show patterns that hold for a while, then break.",{"id":830,"type":825,"conceptId":831,"relation":832,"explanation":833},"conn-data","data-handling","related_to","Counting primes per block, last digits and gaps is data handling: tallies, tables and spotting trends.",{"id":835,"type":50,"title":836,"eyebrow":837,"navLabel":838},"ch11","Findings and a check","Chapter 11","11 Findings",{"id":840,"type":841,"title":842,"terms":843},"gloss-investigate","glossary","Investigation words",[844,848,852,855,859,863,867,871,874,878],{"term":845,"meaning":846,"example":847},"conjecture","A statement that seems true from the evidence but has not been proved.","Twin primes go on forever.",{"term":849,"meaning":850,"example":851},"counterexample","One example that shows a claim is false.","2 + 3 = 5 is a counterexample to \"the sum of two primes is even\".",{"term":853,"meaning":854},"proof","An argument that shows a statement is true in every possible case.",{"term":856,"meaning":857,"example":858},"prime gap","The difference between a prime and the next prime.","The gap after 89 is 8.",{"term":860,"meaning":861,"example":862},"prime desert","A long run of consecutive composite numbers.","90 to 96",{"term":864,"meaning":865,"example":866},"6k ± 1","Numbers one less or one more than a multiple of 6. Every prime above 3 has this form.","29 = 6 × 5 − 1",{"term":868,"meaning":869,"example":870},"highly composite number","A number with more factors than any smaller number. Studied by Ramanujan.","12, 24, 36, 48, 60",{"term":539,"meaning":872,"example":873},"A number made only of the digit 1.","1,111",{"term":875,"meaning":876,"example":877},"remainder class","All the numbers that leave the same remainder when divided by a given number.","Remainder 1 on dividing by 3: 1, 4, 7, 10, …",{"term":879,"meaning":880,"example":881},"necessary condition","Something that must be true, but is not enough on its own.","Ending in 1, 3, 7 or 9 is necessary for a prime above 5.",{"id":883,"type":884,"title":885,"questions":886},"quiz-investigate","quiz","What did the investigations show?",[887,898,911,923,936,949,958,969,982,991],{"itemId":888,"prompt":889,"options":890,"correct":63,"why":897},"prime-and-composite.investigate-q-pi","How many primes are there up to 1,000?",[891,892,893,895],{"id":60,"label":88},{"id":63,"label":94},{"id":66,"label":894},"250",{"id":69,"label":896},"500","A computer count gives 168, compared with 25 up to 100. Primes thin out.",{"itemId":899,"prompt":900,"options":901,"correct":60,"why":910},"prime-and-composite.investigate-q-6col","On a 6-column grid, primes above 3 appear only in the columns under…",[902,904,906,908],{"id":60,"label":903},"1 and 5",{"id":63,"label":905},"1, 3 and 5",{"id":66,"label":907},"2 and 4",{"id":69,"label":909},"any column","The other columns contain only multiples of 2 or 3.",{"itemId":912,"prompt":913,"options":914,"correct":66,"why":922},"prime-and-composite.investigate-q-6k-comp","Which of these is of the form 6k + 1 but is **not** prime?",[915,916,918,920],{"id":60,"label":517},{"id":63,"label":917},"37",{"id":66,"label":919},"49",{"id":69,"label":921},"43","49 = 6 × 8 + 1 = 7 × 7.",{"itemId":924,"prompt":925,"options":926,"correct":63,"why":935},"prime-and-composite.investigate-q-triplet","Why is 3, 5, 7 the only set of three primes each 2 apart?",[927,929,931,933],{"id":60,"label":928},"Nobody has checked big enough numbers",{"id":63,"label":930},"One of n, n + 2, n + 4 is always a multiple of 3",{"id":66,"label":932},"Primes above 7 are all even",{"id":69,"label":934},"It is an unsolved conjecture","The remainders on dividing by 3 cover every case, so one of the three is a multiple of 3; it can only be prime if it is 3.",{"itemId":937,"prompt":938,"options":939,"correct":63,"why":948},"prime-and-composite.investigate-q-euler","n × n + n + 41 is prime for n = 0 to 39. What happens at n = 40?",[940,942,944,946],{"id":60,"label":941},"Prime again",{"id":63,"label":943},"It gives 1,681 = 41 × 41",{"id":66,"label":945},"It gives an even number",{"id":69,"label":947},"It gives 1","40 × 40 + 40 + 41 = 41 × 41 = 1,681. Forty successes do not make a proof.",{"itemId":950,"prompt":951,"options":952,"correct":66,"why":957},"prime-and-composite.investigate-q-middle","The number between the twin primes 101 and 103 is 102. It is a multiple of…",[953,954,955,956],{"id":60,"label":84},{"id":63,"label":360},{"id":66,"label":659},{"id":69,"label":302},"102 = 6 × 17. Every middle number of twins after (3, 5) is a multiple of 6.",{"itemId":959,"prompt":960,"options":961,"correct":60,"why":968},"prime-and-composite.investigate-q-consec-odd","Are two consecutive odd numbers always co-prime?",[962,964,966],{"id":60,"label":963},"Yes",{"id":63,"label":965},"No: 9 and 15 share 3",{"id":66,"label":967},"Only if both are prime","A common factor divides their difference, 2. Both are odd, so the common factor is 1. (9 and 15 are not consecutive odd numbers.)",{"itemId":970,"prompt":971,"options":972,"correct":63,"why":981},"prime-and-composite.investigate-q-desert","2 × 3 × 4 × 5 = 120. Which of these is **guaranteed** composite by the prime-desert trick?",[973,975,977,979],{"id":60,"label":974},"121",{"id":63,"label":976},"123",{"id":66,"label":978},"127",{"id":69,"label":980},"131","123 = 120 + 3, and both parts are divisible by 3. (121 = 11 × 11 happens to be composite too, but the trick does not cover it; 127 and 131 are prime.)",{"itemId":983,"prompt":984,"options":985,"correct":66,"why":990},"prime-and-composite.investigate-q-3factors","Which number has exactly three factors?",[986,987,988,989],{"id":60,"label":349},{"id":63,"label":157},{"id":66,"label":919},{"id":69,"label":88},"49 = 7 × 7 has factors 1, 7, 49. Numbers with three factors are squares of primes.",{"itemId":992,"prompt":993,"options":994,"correct":66,"why":1003},"prime-and-composite.investigate-q-rule-24","A number is divisible by 24 exactly when it is divisible by…",[995,997,999,1001],{"id":60,"label":996},"4 and 6",{"id":63,"label":998},"2 and 12",{"id":66,"label":1000},"3 and 8",{"id":69,"label":1002},"2, 3 and 4","3 and 8 are co-prime with product 24. The others fail for 12 (a, d) or 36 (b).",{"id":1005,"type":1006,"prompt":1007},"reflect-investigate","reflection","Which investigation surprised you most? Write down one pattern you believed at first and then saw break, and one pattern you now think is always true. What would you need to be certain about the second one?",{"id":1009,"type":1010,"title":1011,"points":1012},"cheat-investigate","summary","Findings so far",[1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023],"Primes **thin out**: 25 up to 100, 168 up to 1,000, 78,498 up to a million. But they never run out.","Every prime above 3 is **6k − 1 or 6k + 1**. Not every such number is prime (25, 35, 49, …).","Primes above 5 end in **1, 3, 7 or 9**, roughly equally often.","Twin primes get rarer faster than primes: 35 pairs up to 1,000. Whether they go on forever is **unsolved**.","The number between twin primes (after 3, 5) is a **multiple of 6**.","**3, 5, 7** is the only prime triplet: one of n, n + 2, n + 4 is always a multiple of 3.","Consecutive numbers and consecutive odd numbers are **always co-prime**; consecutive even numbers never are.","Patterns can hold many times and still fail: n × n + n + 41 fails at 40; 2¹¹ − 1 = 23 × 89.","Prime gaps can be as long as you like: 720 + 2 to 720 + 6 are five composites in a row.","Numbers with exactly three factors are **squares of primes**. Number of factors: add 1 to each power and multiply.","Combine divisibility tests only with **co-prime** pieces: 12 = 3 × 4, 24 = 3 × 8.",{"id":1025,"type":1026,"sourceIds":1027},"sources-investigate","sources",[1028,1029,1030,1031,1032,1033,1034],"prime-and-composite-britannica-prime-number","prime-and-composite-britannica-eratosthenes-sieve","prime-and-composite-wiki-twin-prime","prime-and-composite-gimps-mersenne","prime-and-composite-mathisfun-divisibility","prime-and-composite-ncert-class6-prime-time","prime-and-composite-wiki-chebyshev-bias",[1028,1029,1030,1031,1032,1033,1034],"needs_review",{"generatedBy":1038,"notes":1039},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","5c0c9a63c052905eecc2fa7fd486a55dc394df4cb6ec287b6b9d71dc3edbc17e",{"logic:practice":1042,"component:prime-sieve@1":1043,"component:sort-game@1":1044,"component:factor-tree@1":1045,"source:prime-and-composite-britannica-eratosthenes-sieve":1046,"source:prime-and-composite-britannica-prime-number":1047,"source:prime-and-composite-gimps-mersenne":1048,"source:prime-and-composite-mathisfun-divisibility":1049,"source:prime-and-composite-ncert-class6-prime-time":1050,"source:prime-and-composite-wiki-chebyshev-bias":1051,"source:prime-and-composite-wiki-twin-prime":1052},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","76baccc44f3b2ac333537045fea84801ea3c46f2d75f3db8a02355ef0f267041","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","dfb598a254f20e70550d586121be496f7a546ba0254060fd97b95cb919dfd934","af8e7bbdca2e2859b452689299eeef54805896728cef8a361ea1ec0dd63d12d4","8985127013a5a393bbde77d9c7d1e0757013c9dd047aa324fe51fa4706c2edfa","be41b20420652d001b4454ec3ac0d159a0139fb79e77363b0afc6f376ecc405c","d26ef4a1f28e097000ad535bd8d0bb52463c4eb61d815cd4dd383a8db4bac076","f6edb540745600a1873f0a5851335bf576bd243816f181e62085831fd55ac4ce","8f96dca0029308279fd5bc6f2d336d922e43db364b76c46abf2ec0a7b43c9f1f","b0b179308ab5f683db390f67f15856137356f04cbd838b8129c6b89a3c2b8265",{"state":1054,"reviewer":1055,"selfReview":706,"reviewedAt":1056,"method":1057},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899599153]