Problems & Puzzles: Puzzles

Puzzle 89.- The first palprime as a sum of consecutive composites

G. L. Honaker, Jr. asks (30/03/2000) "Can the sum of the first n composite numbers be a palindromic prime?"

Solution

Jud McCranie has produced (31/03/2000) 31 palindromic sums, unfortunately none of these are prime values. This is his largest result:

# of composite = n = 3921264376
composite = 4116490653
sum = 8080970392930790808

The importance of this results is that anyone that decide to search beyond 4116490653 can start his own search from the corresponding sum value calculated by Jud, 8080970392930790808 (the correctness of the Jud's results for me is assured because I have the same 28 first palindromic sums, point in which I stopped)

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Part b) of the puzzle

If we change a bit the puzzle to sum the first  n odd-composites I (CR) have obtained the following (at least one!!!...)  palprime: 9 + 15 + ... + 6321 = 7576757, n = 2339. BTW 2339 is a prime numbers also.

Can you find another one?

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A few hours later, Jud McCranie got the next prime for the part b), and the last <2^32:

9 + 15 + ... + 2110546101 = 1007330577750337001, n= 951894677 ... can you believe it?..951894677  is also a prime number!...

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But...."1007330577750337001 is composite!!!", pointed out Jeff Heleen.

Jud has accepted this fact and has sent a new list of palindromes < 2^32 without any new palprime.

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Jeff Heleen has produced two new results:

part a) n = 6249167674, composite = 6553235945, sum = 20500472188127400502
part b) n = 2822581688, composite = 6224149881,sum = 8806146443446416088

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More results by Jeff are the following (24/05/2000):

For part a there are no further palindromic sums for composites up to 2^34. The final numbers I have if someone wishes to continue are:

n = 16416930072
comp = 17179869184
sum = 141173199339308825420.

For part b, however, there is one more palindromic sum for odd composites < 2^34. This is:

------->         n = 7794689270    <-------
------->   comp = 17109266357   <-------
------->     sum = 66832347055074323866    <-------

with the last sum for those who wish to continue being:

n = 7826995481
comp = 17179869183
sum = 67386223035880684429.

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