Problems & Puzzles: Puzzles Puzzle 9. p1 + p2 + … pk = n ^{m} , m => 2 Let’s define S(Pk) = p1 + p2 + … + pk, namely "the sum of the first k primes" G.L. Honaker, Jr. observes that the sum of the first 9 consecutive prime numbers = 10 ^{2} , and asks "Can k produce other perfect squares, cubes, etc. ?" Judson McCranie has found that S(k) is a perfect square not only for k = 9 but also for k=2474, 6694, 7785 and for … 709838… the current record for perfect squares. Here is the expression of this record : S(P709838) =2 + 3 + 5 + 7 + ... + 10729219 = 1916357^2 Honaker reports that no S(k)perfect cube has been found.
Would you like to beat the current record or to find another perfect power? The current record solution for this old puzzle has been gotten by Giovanni Resta (May 2003):
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