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If a triangular number is even add or subtract 1; if odd, add or subtract 2. Count the number of primes found by subtraction and by addition and it appears that MORE are found by addition than by subtraction. A sample of the first 44 triangular numbers found 17 for subtraction and 26 for addition. Does this difference hold for more triangular numbers considered?

J. M. Bergot
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    A deceptively simple question. – Robert Soupe Aug 21 '18 at 18:02
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    When $n=4m$, the triangular number $(4m)(4m+1)/2=8m^2+2m$ modulo 3 equals 0 or 1. By subtracting one, 1/3 of these numbers are multiples of 3. Such does not happen with adding one. – Marco Aug 21 '18 at 18:10
  • If the triangular number ends in a 6 you cannot subtract 1 and if it ends in a 3, you cannot add 2. One would need a larger sample tested to see if the statistic hold from prediction. – J. M. Bergot Aug 21 '18 at 18:37
  • Only the first even triangle number produces a prime for -1 (6 gives 5), after that the even triangle numbers -1 are never prime. – pietfermat Aug 23 '18 at 09:52
  • Needed certainly is a larger sample as an experiment. Much talk can occur with no solid evidence. Nothing like evidence to get to the crux of the problem. One can see not much reason why subtracting 1 or 2 versus adding 1 or 2 will produce a large discrepancy in the primes produced. – J. M. Bergot Aug 23 '18 at 18:05
  • For subtraction of 1 or 2, 18 out of 60 triangular numbers are forbidden in a three cycles of 20 numbers beginning at T(20) to T(79) for triangular number T(k); for addition of 1 or 2, 16 are forbidden for the same 60 numbers. Hence 30% are forbidden for the first and 23 1/3 for the second; this means the ratio of successes should be 42 to 46, very close. Does experimentation confirm this? – J. M. Bergot Aug 24 '18 at 17:31

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