## Each of a-b, a+n, b+n, a+b+n is a square

To find two numbers,   $a$   and   $b$,   whose difference,   $a - b$   is a square

and such that each,   $a + n$   and   $b + n$,   is a square and their sum,   $a + b + n$,   is a square.

Hint:

Put   $a = x^2 - n$   and   $b = y^2 - n$

then,
$a \; - \; b \; = \; (x^2 - n) \; - \; (y^2 - n) \; = \; x^2 \; - \; y^2$
$a \; + \; b \; + \; n \; = \; (x^2 - n) \; + \; (y^2 - n) \; + \; n \; = \; x^2 + y^2 - n$

$(x^2 - y^2)$   and   $(x^2 + y^2 - n)$   are to be made squares.

math grad - Interest: Number theory
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### 4 Responses to Each of a-b, a+n, b+n, a+b+n is a square

1. David @InfinitelyManic says:

a: 228 b: 84 n: 172 (a-b) 144 (a+n) 400 (b+n) 256 (a+b+n) 484
a: 217 b: 136 n: 8 (a-b) 81 (a+n) 225 (b+n) 144 (a+b+n) 361
a: 216 b: 152 n: 73 (a-b) 64 (a+n) 289 (b+n) 225 (a+b+n) 441
a: 208 b: 64 n: 17 (a-b) 144 (a+n) 225 (b+n) 81 (a+b+n) 289
a: 185 b: 41 n: 215 (a-b) 144 (a+n) 400 (b+n) 256 (a+b+n) 441
a: 180 b: 99 n: 45 (a-b) 81 (a+n) 225 (b+n) 144 (a+b+n) 324
a: 175 b: 111 n: 114 (a-b) 64 (a+n) 289 (b+n) 225 (a+b+n) 400
a: 175 b: 31 n: 50 (a-b) 144 (a+n) 225 (b+n) 81 (a+b+n) 256
a: 145 b: 120 n: 24 (a-b) 25 (a+n) 169 (b+n) 144 (a+b+n) 289
a: 145 b: 64 n: 80 (a-b) 81 (a+n) 225 (b+n) 144 (a+b+n) 289
a: 136 b: 72 n: 153 (a-b) 64 (a+n) 289 (b+n) 225 (a+b+n) 361
a: 112 b: 87 n: 57 (a-b) 25 (a+n) 169 (b+n) 144 (a+b+n) 256
a: 112 b: 31 n: 113 (a-b) 81 (a+n) 225 (b+n) 144 (a+b+n) 256
a: 99 b: 35 n: 190 (a-b) 64 (a+n) 289 (b+n) 225 (a+b+n) 324
a: 85 b: 21 n: 15 (a-b) 64 (a+n) 100 (b+n) 36 (a+b+n) 121
a: 81 b: 56 n: 88 (a-b) 25 (a+n) 169 (b+n) 144 (a+b+n) 225
a: 80 b: 44 n: 20 (a-b) 36 (a+n) 100 (b+n) 64 (a+b+n) 144
a: 57 b: 21 n: 43 (a-b) 36 (a+n) 100 (b+n) 64 (a+b+n) 121
a: 52 b: 27 n: 117 (a-b) 25 (a+n) 169 (b+n) 144 (a+b+n) 196
a: 20 b: 11 n: 5 (a-b) 9 (a+n) 25 (b+n) 16 (a+b+n) 36

2. David @InfinitelyManic says:

0xff < (a,b,n) < 500
a: 385 b: 285 n: 291 (a-b)=100 (a+n)=676 (b+n)=576 (a+b+n)=961
a: 324 b: 275 n: 301 (a-b)=49 (a+n)=625 (b+n)=576 (a+b+n)=900

3. paul says:

Filling a few gaps in with b < a < 500

{20,11,5}… 20 – 11 = 3^2 & 20 + 5 = 5^2 & 11 + 5 = 4^2 & 20 + 11 + 5 = 6^2
{52,27,117}… 52 – 27 = 5^2 & 52 + 117 = 13^2 & 27 + 117 = 12^2 & 52 + 27 + 117 = 14^2
{57,21,43}… 57 – 21 = 6^2 & 57 + 43 = 10^2 & 21 + 43 = 8^2 & 57 + 21 + 43 = 11^2
{80,44,20}… 80 – 44 = 6^2 & 80 + 20 = 10^2 & 44 + 20 = 8^2 & 80 + 44 + 20 = 12^2
{81,56,88}… 81 – 56 = 5^2 & 81 + 88 = 13^2 & 56 + 88 = 12^2 & 81 + 56 + 88 = 15^2
{85,21,15}… 85 – 21 = 8^2 & 85 + 15 = 10^2 & 21 + 15 = 6^2 & 85 + 21 + 15 = 11^2
{99,35,190}… 99 – 35 = 8^2 & 99 + 190 = 17^2 & 35 + 190 = 15^2 & 99 + 35 + 190 = 18^2
{112,31,113}… 112 – 31 = 9^2 & 112 + 113 = 15^2 & 31 + 113 = 12^2 & 112 + 31 + 113 = 16^2
{112,87,57}… 112 – 87 = 5^2 & 112 + 57 = 13^2 & 87 + 57 = 12^2 & 112 + 87 + 57 = 16^2
{136,72,153}… 136 – 72 = 8^2 & 136 + 153 = 17^2 & 72 + 153 = 15^2 & 136 + 72 + 153 = 19^2
{145,64,80}… 145 – 64 = 9^2 & 145 + 80 = 15^2 & 64 + 80 = 12^2 & 145 + 64 + 80 = 17^2
{145,120,24}… 145 – 120 = 5^2 & 145 + 24 = 13^2 & 120 + 24 = 12^2 & 145 + 120 + 24 = 17^2
{153,104,472}… 153 – 104 = 7^2 & 153 + 472 = 25^2 & 104 + 472 = 24^2 & 153 + 104 + 472 = 27^2
{175,31,50}… 175 – 31 = 12^2 & 175 + 50 = 15^2 & 31 + 50 = 9^2 & 175 + 31 + 50 = 16^2
{175,111,114}… 175 – 111 = 8^2 & 175 + 114 = 17^2 & 111 + 114 = 15^2 & 175 + 111 + 114 = 20^2
{180,99,45}… 180 – 99 = 9^2 & 180 + 45 = 15^2 & 99 + 45 = 12^2 & 180 + 99 + 45 = 18^2
{185,41,215}… 185 – 41 = 12^2 & 185 + 215 = 20^2 & 41 + 215 = 16^2 & 185 + 41 + 215 = 21^2
{208,64,17}… 208 – 64 = 12^2 & 208 + 17 = 15^2 & 64 + 17 = 9^2 & 208 + 64 + 17 = 17^2
{208,108,468}… 208 – 108 = 10^2 & 208 + 468 = 26^2 & 108 + 468 = 24^2 & 208 + 108 + 468 = 28^2
{208,159,417}… 208 – 159 = 7^2 & 208 + 417 = 25^2 & 159 + 417 = 24^2 & 208 + 159 + 417 = 28^2
{216,152,73}… 216 – 152 = 8^2 & 216 + 73 = 17^2 & 152 + 73 = 15^2 & 216 + 152 + 73 = 21^2
{217,136,8}… 217 – 136 = 9^2 & 217 + 8 = 15^2 & 136 + 8 = 12^2 & 217 + 136 + 8 = 19^2
{228,84,172}… 228 – 84 = 12^2 & 228 + 172 = 20^2 & 84 + 172 = 16^2 & 228 + 84 + 172 = 22^2
{259,195,30}… 259 – 195 = 8^2 & 259 + 30 = 17^2 & 195 + 30 = 15^2 & 259 + 195 + 30 = 22^2
{260,35,29}… 260 – 35 = 15^2 & 260 + 29 = 17^2 & 35 + 29 = 8^2 & 260 + 35 + 29 = 18^2
{265,165,411}… 265 – 165 = 10^2 & 265 + 411 = 26^2 & 165 + 411 = 24^2 & 265 + 165 + 411 = 29^2
{265,216,360}… 265 – 216 = 7^2 & 265 + 360 = 25^2 & 216 + 360 = 24^2 & 265 + 216 + 360 = 29^2
{273,129,127}… 273 – 129 = 12^2 & 273 + 127 = 20^2 & 129 + 127 = 16^2 & 273 + 129 + 127 = 23^2
{276,51,349}… 276 – 51 = 15^2 & 276 + 349 = 25^2 & 51 + 349 = 20^2 & 276 + 51 + 349 = 26^2
{297,41,103}… 297 – 41 = 16^2 & 297 + 103 = 20^2 & 41 + 103 = 12^2 & 297 + 41 + 103 = 21^2
{320,176,80}… 320 – 176 = 12^2 & 320 + 80 = 20^2 & 176 + 80 = 16^2 & 320 + 176 + 80 = 24^2
{324,224,352}… 324 – 224 = 10^2 & 324 + 352 = 26^2 & 224 + 352 = 24^2 & 324 + 224 + 352 = 30^2
{324,275,301}… 324 – 275 = 7^2 & 324 + 301 = 25^2 & 275 + 301 = 24^2 & 324 + 275 + 301 = 30^2
{329,104,296}… 329 – 104 = 15^2 & 329 + 296 = 25^2 & 104 + 296 = 20^2 & 329 + 104 + 296 = 27^2
{340,84,60}… 340 – 84 = 16^2 & 340 + 60 = 20^2 & 84 + 60 = 12^2 & 340 + 84 + 60 = 22^2
{369,225,31}… 369 – 225 = 12^2 & 369 + 31 = 20^2 & 225 + 31 = 16^2 & 369 + 225 + 31 = 25^2
{384,159,241}… 384 – 159 = 15^2 & 384 + 241 = 25^2 & 159 + 241 = 20^2 & 384 + 159 + 241 = 28^2
{385,129,15}… 385 – 129 = 16^2 & 385 + 15 = 20^2 & 129 + 15 = 12^2 & 385 + 129 + 15 = 23^2
{385,285,291}… 385 – 285 = 10^2 & 385 + 291 = 26^2 & 285 + 291 = 24^2 & 385 + 285 + 291 = 31^2
{385,336,240}… 385 – 336 = 7^2 & 385 + 240 = 25^2 & 336 + 240 = 24^2 & 385 + 336 + 240 = 31^2
{441,216,184}… 441 – 216 = 15^2 & 441 + 184 = 25^2 & 216 + 184 = 20^2 & 441 + 216 + 184 = 29^2
{448,124,452}… 448 – 124 = 18^2 & 448 + 452 = 30^2 & 124 + 452 = 24^2 & 448 + 124 + 452 = 32^2
{448,348,228}… 448 – 348 = 10^2 & 448 + 228 = 26^2 & 348 + 228 = 24^2 & 448 + 348 + 228 = 32^2
{448,399,177}… 448 – 399 = 7^2 & 448 + 177 = 25^2 & 399 + 177 = 24^2 & 448 + 399 + 177 = 32^2
{451,51,174}… 451 – 51 = 20^2 & 451 + 174 = 25^2 & 51 + 174 = 15^2 & 451 + 51 + 174 = 26^2
{459,59,382}… 459 – 59 = 20^2 & 459 + 382 = 29^2 & 59 + 382 = 21^2 & 459 + 59 + 382 = 30^2
{500,59,341}… 500 – 59 = 21^2 & 500 + 341 = 29^2 & 59 + 341 = 20^2 & 500 + 59 + 341 = 30^2
{500,275,125}… 500 – 275 = 15^2 & 500 + 125 = 25^2 & 275 + 125 = 20^2 & 500 + 275 + 125 = 30^2

Paul.

• benvitalis says:

Correct. Thanks to David and Paul for taking the time to solve this