A396526 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 71^2)^2 = y^2.

0, 923, 2040, 8379, 11360, 15123, 19880, 25899, 61820, 79023, 100820, 128439, 163436, 372623, 472860, 599879, 760836, 964799, 2184000, 2768219, 3508536, 4446659, 5635440, 12741459, 16146536, 20461419, 25929200, 32857923, 74274836, 94121079, 119270060

Offset

1,2

Comments

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2m^2 - 1 a (prime) number in A066436, m >= 4 (p >= 31), the first five consecutive solutions are (0, p^2), (4m^3+2m^2-2m-1, 4m^4+4m^3-2m-1), (8m^3+8m^2+4m, 4m^4+8m^3+12m^2+4m+1), (12m^4-40m^3+44m^2-20m+3, 20m^4-56m^3+60m^2-28m+5), (12m^4-20m^3+2m^2+10m-4, 20m^4-28m^3+14m-5) and the other solutions are defined by (X(n), Y(n)) = (3*X(n-5) + 2*Y(n-5) + p^2, 4*X(n-5) + 3*Y(n-5) + 2p^2).

X(n) = 6*X(n-5) - X(n-10) + 2p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2p^2).

Links

Table of n, a(n) for n=1..31.

Formula

a(n) = 6*a(n-5) - a(n-10) + 10082 for n >= 11; a(1)=0, a(2)=923, a(3)=2040, a(4)=8379, a(5)=11360, a(6)=15123, a(7)=19880, a(8)=25899, a(9)=61820, a(10)=79023.

Example

For p=71 (m=6) the first five (5) consecutive solutions are (0, 5041), (923, 6035), (2040, 7369), (8379, 15821), (11360, 19951)

Crossrefs

Cf. A066436 (primes of the form 2*m^2 - 1).

Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), A331265 (p=31), A332000 (p=47), this sequence (p=71), A396534 (p=79).

Sequence in context: A235881 A051984 A232732 * A228673 A365469 A268849

Adjacent sequences: A396523 A396524 A396525 * A396527 A396528 A396530

Keyword

nonn

Author

Mohamed Bouhamida, May 28 2026

Status

approved