To find positive integers such that

, and

Note that

The sequence for the values of ** x** : 2, 5, 10, 17, 26

and

2

5 – 2 = 3

10 – 5 = 5

17 – 10 = 7

26 – 17 = 9

…………

…………??

Does the sequence continue?

Yes, it does.

—————————————————-

Paul found:

The generating functions for x and y are:

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Yes it does. The generating functions for x and y are:-

n^2+1 and n^2(n^2+1)

These then form the following generating functions for the squares of

x + y = (1 + n^2)^2 and

x y = (n + n^3)^2

Here are the next 10 in the sequence format {x, y, a^2, b^2}

37, 1332, 1369, 49284

50, 2450, 2500, 122500

65, 4160, 4225, 270400

82, 6642, 6724, 544644

101, 10100, 10201, 1020100

122, 14762, 14884, 1800964

145, 20880, 21025, 3027600

170, 28730, 28900, 4884100

197, 38612, 38809, 7606564

226, 50850, 51076, 11492100

Paul.