Let's define S(k) = p₁ + p₂ + ... + p_k, i.e. "the sum of the first k primes"
The sum of the first 9 consecutive prime numbers equals 100, a square number.

Find all k (below 9999) that will similarly produce perfect squares.

Extended just far enough to get one more example beyond the range asked for:

   10   for I=1 to 999999
   20      P=nxtprm(P):Sum=Sum+P
   30      Sr=int(sqrt(Sum)+0.5)
   40      if Sr*Sr=Sum then print I,P,Sum,Sr
   50   next

finds

    k                p(k)               S(k)     sqrt(S(k))
    9                  23                  100          10
 2474               22073             25633969        5063
 6694               67187            212372329       14573
 7785               79427            292341604       17098
 709838          10729219        3672424151449     1916357

Posted by Charlie on 2012-06-29 16:04:50