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While having lunch in our cafeteria, some mathematicians told me of a quite interesting problem:

There are infinitely many numbers that can't be written as a sum of a prime and a triangular number.

They've said that they all failed to prove that theorem. Unfortunately, I failed in proving that as well. Does someone of you know a proof of that? Or is the theorem false at all?

The triangular numbers are given in explicit form as $T_n = \frac{n(n+1)}{2}$.

user26857
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MathJoker
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  • The difference of two triangular numbers is usually not prime, so usuallt you cannot write a triangular number in this form. You may or may not want to modify your question to exclude that type of response. Also see http://math.stackexchange.com/questions/1586507/ for this idea. – quid Jul 29 '16 at 10:00
  • In order to improve the quality of the question (which is generally interesting IMO), you can: 1. Include the mathematical definition of triangular number (instead of leaving it for the reader to google). 2. Give an example of at least one such number (since it is not that easy to find). – barak manos Jul 29 '16 at 10:25
  • @quid: Thank you for the link, but I can't see why the difference of two triangular numbers is usually not prime. – MathJoker Jul 29 '16 at 13:46
  • @barak manos: Thank you for your advice. In the link posted by quid, one shows that 2016 can't be written as a sum of a prime and a triangular number. – MathJoker Jul 29 '16 at 13:48

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Take $n < m$. Notice that $d = m(m+1)/2 - n(n+1)/2 = (1/2)(m-n)(m+n+1)$. Say $m > 2$ is even.

If $n < m - 1$ is odd, then $m + n + 1 > 4$ is even, and so $d$ is composite. If $n < m - 2$ is even, then $d$ will have non-trivial factors $(1/2)(m-n)$ and $m + n + 1$.

This leaves $n = m - 1$, yielding $d = m$, or $n = m - 2$, yielding $d = 2m - 1$. So choose $m$ such that $2m - 1$ is not prime, and the numbers $T_m$ fit the criterion. For instance, $m = 6k + 2$ always works, $k \ge 1$.

Mr. Chip
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  • m need not be even, only prime. 30k + 2, 3, 5, 8, 12, 14, 18, 20, 25, 26, 28 will work for k ≥ 1. Now the difficult question: Are there infinitely many non-triangular numbers that are not the sum of a triangular number and a prime? – gnasher729 Jul 29 '16 at 15:59
  • @gnasher729 there is a conjecture to the contrary by Z.W. Sun.See conj 1.1 in https://arxiv.org/abs/0803.3737 – quid Jul 29 '16 at 18:32
  • After some experimentation, I'd suggest that the integers which are not the sum of a triangular number and a prime are the triangular numbers Tn where neither n nor 2n-1 are prime, plus the numbers 2, 7, 61, 211, and 216. For example, for x = 3,247,410,348 x - Tn is composite for a bit over 1,100 of the largest triangular numbers less than x, but x - Tn would have to be composite for over 80,000 triangular numbers. – gnasher729 Jul 29 '16 at 18:43