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Math Help - Sum of prime numbers

  1. #1
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    Sum of prime numbers

    What is the sum of all prime numbers p such that : 8p+120 is a triangle number. A triangle number is a number in the form (n(n+1))/2.

    Any hints? Thanx.
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  2. #2
    Senior Member Paze's Avatar
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    Re: Sum of prime numbers

    Was googling your problem and came across this:

    It turns out that there are only 2 primes that
    satisfy 8p+120 = a triangular number.
    Clearly, p = 2 works:
    16+120 = 136 = 16*17/2.
    We will show that the only other value of p
    that works is p = 47.
    Thus the sum of all the required primes is 49.
    We need the following result:
    If n is a triangular number then 8n+1 is a square.
    So we can write the condition of the problem as
    64p + 961 = x^2
    or
    (x+31)(x-31) = 64p.
    Suppose p = 2
    Since x > 31 we have x -31 = 2
    x = 33, 128 + 961 = 1089 = 33^2.
    So let's assume p is an odd prime
    Then, because p is prime, we must have
    one of the following:
    a. x-31 = 32, x+31 = 2p
    Then x = 63, 63+31 = 94
    94*32 + 961 = 3969, a square.
    So p = 47. 8p+120 = 496 = 31*32/2.
    b. x-31 = 16, x+31 = 4p
    x = 47. 47+31 = 78, but
    78 is not of the form 4p.
    c). x-31 = 8. x+31 = 8p
    x = 39, 39+31 = 70,
    but 70 is not of the form 8p.
    Similarly, the remaining 2 cases are impossible.
    The answer to the problem is therefore 49.
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