A369498 Integers k such that k = a^2 + b^2 = c^2 + d^2 and a + b = 3(c - d), where a, b, c and d are distinct positive integers.

65, 260, 585, 650, 1037, 1040, 1625, 1853, 2340, 2378, 2465, 2600, 3185, 3650, 4148, 4160, 5265, 5513, 5850, 6500, 6890, 7298, 7412, 7865, 8177, 9333, 9360, 9512, 9593, 9860, 10400, 10985, 12740, 14600, 14625, 14690, 16133, 16250, 16592, 16640, 16677, 18005

Offset: 1

Author

Gonzalo Martínez, Jan 24 2024

Keywords

• nonn

Comments

These numbers allow the generation of infinitely many arithmetic progressions (A.P.) of length 4 whose elements belong to A000404.

For example, (37, 61, 85, 109) is an A.P. whose difference is 24, and 37, 61, 85 and 109 are in A000404.

To prove that in A000404 there exist infinitely many 4-tuples (x,y,z,w) that form an A.P. we can find a 4-tuple as a function of a parameter m. For this purpose, we consider the following expressions:

x = (m - a)^2 + (m - b)^2 = 2m^2 - 2m(a + b) + a^2 + b^2

y = (m - c)^2 + (m + d)^2 = 2m^2 - 2m(c - d) + c^2 + d^2

z = (m + c)^2 + (m - d)^2 = 2m^2 + 2m(c - d) + c^2 + d^2

w = (m + a)^2 + (m + b)^2 = 2m^2 + 2m(a + b) + a^2 + b^2

where a, b, c, d are distinct integers such that a^2 + b^2 = c^2 + d^2. Therefore, x, y, z, w will be in A.P. if x + z = 2y, whence we conclude that a + b = 3(c-d).

Thus, if k = a^2 + b^2 = c^2 + d^2 and a + b = 3(c - d), where a, b, c and d are distinct positive integers, then (x, y, z, w) form an A.P. for all positive integers m, and if m > min{a,b,c,d} then all elements belong to A000404.

The smallest number with this property is 65, since 65 = 8^2 + 1^2 = 7^2 + 4^2 and 8 + 1 = 3*(7 - 4). Taking k = 65 and m = 2, the tuple (37, 61, 85, 109) results.

Examples

			1037 is a term because 1037 = 26^2 + 19^2 = 29^2 + 14^2 and 26 + 19 = 3*(29 - 14).
		

Links

• D. R. Heath-Brown, Linear relations amongst sums of two squares, London Mathematical Society Lecture Note Series, 303 (2003), 133 - 176.

Crossrefs

Cf. A000404, A007692.

Programs

Python

from math import isqrt
def A369498_list(n):
    return sorted([
        a**2 + b**2
        for a in range(1, isqrt(n) + 1)
        for b in range(1, a)
        for c in range(1, isqrt(n) + 1)
        for d in range(1, c)
        if a != c and a != d
        and a**2 + b**2 == c**2 + d**2
        and a + b == 3 * (c - d)
        and a**2 + b**2 <= n
    ])
print(A369498_list(18500))