Primes not expressible as the sum of two triangular numbers.

[Davide Rotondo]

A117382

Primes not expressible as the sum of two triangular numbers.

5, 17, 19, 23, 41, 47, 53, 59, 71, 89, 103, 107, 109, 113, 131, 149, 167, 173, 179, 197, 223, 229, 233, 239, 251, 257, 269, 271, 283, 293, 311, 313, 317, 337, 347, 349, 359, 383, 397, 401, 419, 431, 439, 449, 457, 467, 479, 491, 503, 509, 521, 523, 547, 557

Today I controlled the sequence

A094524

Primes of form 3*prime(m) + 2.

11, 17, 23, 41, 53, 59, 71, 89, 113, 131, 179, 239, 251, 269, 293, 311, 383, 419, 449, 491, 503, 521, 593, 599, 683, 701, 719, 773, 809, 881, 941, 953, 1013, 1049, 1061, 1103, 1151, 1193, 1229, 1259, 1301, 1319, 1373, 1439, 1499, 1511, 1571, 1709, 1733, ...

and the sequence

A136082

Son primes of order 5.

3, 11, 17, 23, 41, 53, 59, 107, 131, 167, 173, 179, 191, 257, 263, 269, 389, 401, 431, 461, 467, 479, 521, 563, 569, 599, 647, 653, 677, 683, 719, 773, 821, 839, 857, 887, 947, 971, 1031, 1049, 1061, 1091, 1103, 1151, 1181, 1217, 1223, 1259, 1277, 1301

and I noticed that except for 3 and 11 all the primes belonging to these sequences are part of A117382

Primes not expressible as the sum of two triangular numbers.

Is this true?

See you soon

Davide

[Davide Rotondo]

Just now,I check the firs 100  Primes not expressible as the sum of two triangular numbers using a list that I a made with the help of chatgpt and I don't know if it is correct; however my check for Son primes of order 5 fail, while the cechk for Primes of form 3*prime(m) + 2 still resist. I hope.

See you soon

Davide

[D. S. McNeil]

I think your claim holds for A094524. n is a sum of two triangulars iff every prime 3 mod 4 in the factorization of 8n+2 has an even exponent, via the two squares sum theorem and some arithmetic. Triangulars are T_x=x(x+1)/2, so 4 T_x + 1 = (2x + 1)^2, and so n = T_x + T_y leads to 8 n + 2 = (2x+1)^2 + (2y+1)^2. So 8n+2 needs to be the sum of two odd squares, and the two squares sum theorem (https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem) applies. Now let q=3*p+2 be prime (with p prime). Then 8q+2=8*(3p+2)+2=24p+16+2=24p+18=6*(4p+3)=2 * 3^1 * (4p+3)^1. We check factors of 3 for the two squares constraint. For q to be the sum of two triangulars, 3 must divide 4p+3 an odd number of times to make the total number of factors of 3 even. So we need at least (4p + 3) mod 3 = 0, or p mod 3 =0, but the only prime 0 mod 3 is 3. This would give 2*3*(15)=2*5*3^2, which works, as the only prime 3 mod 4 has an even exponent. So the only q expressible as the sum of two triangular numbers has p=3, for q=3*3+2=11. Every other p leads to a q we can't express. In other words, the only element of A117382 not in A094524 is 11, T(1)+T(4). Doug #precoffeebugslikely

[Davide Rotondo]

Thank you very much Doug!

See you soon

Davide

Il giorno ven 3 ott 2025 alle 1:28 PM D. S. McNeil <dsm...@gmail.com> ha scritto:

I think your claim holds for A094524. n is a sum of two triangulars iff every prime 3 mod 4 in the factorization of 8n+2 has an even exponent, via the two squares sum theorem and some arithmetic. Triangulars are T_x=x(x+1)/2, so 4 T_x + 1 = (2x + 1)^2, and so n = T_x + T_y leads to 8 n + 2 = (2x+1)^2 + (2y+1)^2. So 8n+2 needs to be the sum of two odd squares, and the two squares sum theorem (https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem) applies. Now let q=3*p+2 be prime (with p prime). Then 8q+2=8*(3p+2)+2=24p+16+2=24p+18=6*(4p+3)=2 * 3^1 * (4p+3)^1. We check factors of 3 for the two squares constraint. For q to be the sum of two triangulars, 3 must divide 4p+3 an odd number of times to make the total number of factors of 3 even. So we need at least (4p + 3) mod 3 = 0, or p mod 3 =0, but the only prime 0 mod 3 is 3. This would give 2*3*(15)=2*5*3^2, which works, as the only prime 3 mod 4 has an even exponent. So the only q expressible as the sum of two triangular numbers has p=3, for q=3*3+2=11. Every other p leads to a q we can't express. In other words, the only element of A117382 not in A094524 is 11, T(1)+T(4). Doug #precoffeebugslikely

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