## Num3er 725

Find an integer   $N$   which has 3 representations as a sum of 2 squares,

$N \; = \; a_1^2 \; + \; b_1^2 \; = \; a_2^2 \; + \; b_2^2 \; = \; a_3^2 \; + \; b_3^2$

such that,

$S_1 \; = \; a_1 \; + \; b_1$
$S_2 \; = \; a_2 \; + \; b_2$
$S_3 \; = \; a_3 \; + \; b_3$

$S_1, \; S_2$,   and   $S_3$   are in arithmetic progression.

For example,

$725 \; = \; 7^2 \; + \; 26^2 \; = \; 10^2 \; + \; 25^2 \; = \; 14^2 \; + \; 23^2$

$7 \; + \; 26 \; = \; 33$
$10 \; + \; 25 \; = \; 35$
$14 \; + \; 23 \; = \; 37$

Paul found :

math grad - Interest: Number theory
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### 2 Responses to Num3er 725

1. Paul says:

Here is a list with N<=1000, Format is N, {{a1, b1},{a2,b2},{a3,b3}},{s1, s2, s3}

425 , {{5,20},{8,19},{13,16}} , {25,27,29}
725 , {{7,26},{10,25},{14,23}} , {33,35,37}
1025 , {{1,32},{8,31},{20,25}} , {33,39,45}
1625 , {{16,37},{20,35},{28,29}} , {53,55,57}
1700 , {{10,40},{16,38},{26,32}} , {50,54,58}
2125 , {{3,46},{10,45},{19,42}} , {49,55,61}
2210 , {{19,43},{23,41},{29,37}} , {62,64,66}
2465 , {{8,49},{16,47},{28,41}} , {57,63,69}
2900 , {{14,52},{20,50},{28,46}} , {66,70,74}
3445 , {{14,57},{23,54},{41,42}} , {71,77,83}
3770 , {{7,61},{17,59},{31,53}} , {68,76,84}
3825 , {{15,60},{24,57},{39,48}} , {75,81,87}
3965 , {{11,62},{22,59},{43,46}} , {73,81,89}
4100 , {{2,64},{16,62},{40,50}} , {66,78,90}
4225 , {{16,63},{25,60},{39,52}} , {79,85,91}
4505 , {{28,61},{32,59},{37,56}} , {89,91,93}
4930 , {{13,69},{21,67},{31,63}} , {82,88,94}
5330 , {{1,73},{17,71},{43,59}} , {74,88,102}
5525 , {{7,74},{14,73},{22,71}} , {81,87,93}
5525 , {{7,74},{22,71},{50,55}} , {81,93,105}
5525 , {{14,73},{25,70},{41,62}} , {87,95,103}
6290 , {{19,77},{31,73},{53,59}} , {96,104,112}
6305 , {{8,79},{23,76},{47,64}} , {87,99,111}
6500 , {{32,74},{40,70},{56,58}} , {106,110,114}
6525 , {{21,78},{30,75},{42,69}} , {99,105,111}
6800 , {{20,80},{32,76},{52,64}} , {100,108,116}
7565 , {{13,86},{29,82},{61,62}} , {99,111,123}
7625 , {{35,80},{43,76},{56,67}} , {115,119,123}
7685 , {{31,82},{38,79},{47,74}} , {113,117,121}
8177 , {{44,79},{49,76},{56,71}} , {123,125,127}
8450 , {{13,91},{23,89},{35,85}} , {104,112,120}
8500 , {{6,92},{20,90},{38,84}} , {98,110,122}
8585 , {{11,92},{29,88},{64,67}} , {103,117,131}
8840 , {{38,86},{46,82},{58,74}} , {124,128,132}
9225 , {{3,96},{24,93},{60,75}} , {99,117,135}
9425 , {{4,97},{20,95},{41,88}} , {101,115,129}
9425 , {{31,92},{41,88},{55,80}} , {123,129,135}
9860 , {{16,98},{32,94},{56,82}} , {114,126,138}

Paul.

• benvitalis says:

Cool! in this list $d \; = \; S_2 \; - \; S_1 \; = \; S_3 \; - \; S_2$ is always an even number. Can you find an $N$ such that d is odd?