Problems & Puzzles: Puzzles Puzzle 18. Some special sums of consecutive primes :
I have found only two sum of this types : 2 + 3 = 5 ; S(p_{1}>p_{2}) = S(p_{3}>p_{3 }) and 2 + 3 + … 3833 = 3847 + … + 5557 ; S(p_{1}>p_{532}) = S(p_{533}>p_{733 }) Find 3 more examples
p_{k = }5, 71, 369119, 415074643 Find three k values more. *** At 31/08/98 Jud McCranie informs the following: part 1: "no more solutions for L <
57,442,974 (P_{L} = 1,137,118,693)" Later (7/7/2000) he added to part 2: "No others p < 29,505,444,491 (sum < 2^64)... but see A007506". According to that sequence it results that the part 2 was established time ago (?) by Robert G. Wilson. *** By obliviousness I (C.R.) tackled (24/2/2001) again the question 1, but this time I recorded all the solutions such that S(p_{1}>p_{K})  S(p_{K+1}>p_{L }) <10. Thanks to that missing I discover the following two almost solution: 1+S(2 >23117) = S(23131 > 33359) *** Giovanni Resta found (May 2003) the following surprising solution for the question 1:
***





