Problems & Puzzles: Puzzles

 Puzzle 1004. Sum of all the first primes 4k+3 as perfect squares This last week, Carlos Rivera, was trying to find perfect squares as sum of the first primes 4k+3, starting in 3.   S=3+7+11+...Pn = X^2.................(1)   Unexpectedly he found no one single solution to (1) adding primes until Pn< 2^32.   Q1. Can you try to find the smallest solution to (1) or prove that this target is impossible?   Is there any particular issue to start the sum S with the prime 3?   Because if we start with any other prime P1, is commonly easy to find perfect square solutions. Examples:   P1      Pn        S                   X  7       93059 197993041         14071  11      211    2401                  49  19      67      289                   17  23      47      144                   12  31      47      121                   11  43      4519   643204              802   Q2. Starting with 3 makes impossible to get solutions? Why?

During the week 6-12 June, 2020, contributions came from Giovanni Resta, Oscar Volpatti, Emmanuel Vantieghem.

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Giovanni wrote:

3 + 7 + 11 + ... + 1486287894323 = 141665900839^2

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Oscar wrote:

I don't think that there is any particular issue to start the sum S with the prime 3.
If we check the first 100 primes P1 = 4k+3 as starting points, we find solutions with Pn < 2^32 for only 90 of them.
An exhaustive search up to Pn < 1.7e12 provides solutions for 6 more primes:

491 + ... + 9693570443 = 1022179976^2;
463 + ... + 16660213963 = 1736015962^2;
1051 + ... + 32181040607 = 3306335314^2;
383 + ... + 45512443063 = 4642150049^2;
971 + ... + 394053373223 = 38499816192^2;
3 + ... + 1486287894323 = 141665900839^2.

So the prime 3 is just "unlucky", but there are "unluckier" primes, like 103, 139, 227, and 1091.

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Emmanuel wrote:

I computed the sums of the first primes of the form  4m+3  until  I got  896904502  numbers.
The last prime being 42000000139 which gave the sum  18413351299021328182.
But I did not meet any square.
The probability that one of these sums would be square is roughly
896904502 / Sqrt( 18413351299021328182) = +/- 0.209.
I think the probability will decrease slowly when I continu.
So, I think one day (with a little bit of luck) we''l meet a square.

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