mersenneforum.org Numbered Triangle: (lack of)Prime Patterns
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 2007-08-01, 04:01 #1 Stanek   24·5·59 Posts Numbered Triangle: (lack of)Prime Patterns My first post here, Hi All. Assume we arrange numbers like this. Code:  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 There appears to be a pattern between diagonal columns that can contain prime numbers to those that never will. The 2,3,5,7 are anomalies because no primes exist past these numbers. I created a script to quickly check for primes down to the billionth row to satisfy my definition of no primes in these diagonal rows. 1 3-never 0 prime rows in between 2 5-never 1 prime rows in between 4 8-primes 7 12-never 2 prime rows in between 11 17-primes 16 23-primes 22 30-never 3 prime rows in between 29 38-primes 37 47-primes 46 57-primes 56 68-never 4 prime rows in between 67 80-primes 79 93-primes 92 107-primes 106 122-primes 121 138-never 5 prime rows in between The gap between a diagonal column never containing a prime starts at 0 and increases by one. I've tested this for a few thousand rows and it holds true. My question is why? What is happening here? Obviously I assume this has to do with ordering the numbers in such a fashion. But for the life of me I cannot understand why this is happening. Any nudge in the right direction would be appreciated. Thanks
2007-08-01, 14:52   #2
wblipp

"William"
May 2003
New Haven

23·103 Posts

Quote:
 Originally Posted by Stanek Any nudge in the right direction would be appreciated.
Find the quadratic expression for each of your diagonals (hint - the far right diagonal is n(n+1)/2, the others are offset from this.)

Then try factoring the expression with the quadratic formula.

William

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