Problems & Puzzles:
Puzzles
Puzzle 18. Some special sums of consecutive primes :
 S(p_{1}>p_{K}) = S(p_{K+1}>p_{L })
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
 I have found that S(pk)@ pk = 0
for :
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)"
part 2: no more solutions for primes < 2^32.".
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)
S(2 > 8358529) = S( 8358563 > 11956103) +1
***
Giovanni Resta found (May
2003) the following surprising solution for the question 1:
I would like to submit a new solution for Puzzle 18,
"Some special sums of consecutive primes" In particular about the problem
18.1: p_1 + ...+ p_k = p_(k+1) + .... + p_h whose largest solution was
2+3+...3833 = 3847+...+5557 (k=532, h=733)
I found this new solution:
for k = 18151265107 and h = 25492021989, that is:
2+3+... + 468872968241 = 468872968243+ ...+
667515565537 = 4169395490114624428834
***
On July 5, 2023, Paul W. Dyson wrote:
Last year I ran a program for
about a month and a half on a RTX3060Ti GPU to find the next
solution for S(pk) % pk = 0. This is Puzzle 18 question 2 on your
Prime Puzzles website. The next value is pk
= 55,691,042,365,834,801. This sequence is OEIS A007506. (See also
A024011, A028581 and A028582)
The code was also searching
for the next solution of S(pk) % k = 0 (i.e. % k, rather than %
pk). So I kept it running until I found a solution to that one
too. After a total of about 5 and a half months it found k =
6,361,476,515,268,337. This sequence is OEIS A045345.
I kept looking for S(pk) % pk = 0 for the whole time, and found that
the next solution must be greater than pk = 253,814,097,223,614,463.
...
I've just seen that S(pk) % k = 0 is Puzzle 31. So I have a
solution to that puzzle as well.
***
