## z^2 – y^2 = x^2 – c^2 = w^2 – b^2 = d^2 – a^2

Find distinct positive integers   $a, b, c, d, w, x, y, z$,   such that

$z^2 - y^2 = x^2 - c^2 = w^2 - b^2 = d^2 - a^2$

and

$c^2 - a^2 = y^2 - w^2$

For example,

(1)

z = 109   …..   y = 89   ….. x = 81   …..   c = 51
w = 73   ……   b = 37   ….. d = 63   …..   a = 3

$109^2 - 89^2 = 81^2 - 51^2 = 73^2 - 37^2 = 63^2 - 3^2 = 3960$

$51^2 - 3^2 = 89^2 - 73^2 = 2592$

(2)

z = 97   …..   y = 83   …..   x = 79   …..   c = 61
w = 57   …..   b = 27   …..   d = 51   …..   a = 9

$97^2 - 83^2 = 79^2 - 61^2 = 57^2 - 27^2 = 51^2 - 9^2 = 2520$

$61^2 - 9^2 = 83^2 - 57^2 = 3640$

(3)

z = 334   …..   y = 326   …..   x = 226   …..   c = 214
w = 269   …..   b = 259   …..   d = 131   …..   a = 109

$334^2 - 326^2 = 226^2 - 214^2 = 269^2 - 259^2 = 131^2 - 109^2 = 5280$

$214^2 - 109^2 = 326^2 - 269^2 = 33915$

Find other solutions

math grad - Interest: Number theory
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### 2 Responses to z^2 – y^2 = x^2 – c^2 = w^2 – b^2 = d^2 – a^2

1. Paul says:

Here are a few more

z = 173 … y = 157 … x = 131 … c = 109
w = 122 … b = 98 … d = 86 … a = 46
173^2 – 157^2 = 131^2 – 109^2 = 122^2 – 98^2 = 86^2 – 46^2 = 5280
109^2 – 46^2 = 157^2 – 122^2 = 9765

z = 334 … y = 326 … x = 269 … c = 259
w = 226 … b = 214 … d = 131 … a = 109
334^2 – 326^2 = 269^2 – 259^2 = 226^2 – 214^2 = 131^2 – 109^2 = 5280
259^2 – 109^2 = 326^2 – 226^2 = 55200

z = 218 … y = 202 … x = 178 … c = 158
w = 134 … b = 106 … d = 94 … a = 46
218^2 – 202^2 = 178^2 – 158^2 = 134^2 – 106^2 = 94^2 – 46^2 = 6720
158^2 – 46^2 = 202^2 – 134^2 = 22848

z = 238 … y = 218 … x = 202 … c = 178
w = 134 … b = 94 … d = 106 … a = 46
238^2 – 218^2 = 202^2 – 178^2 = 134^2 – 94^2 = 106^2 – 46^2 = 9120
178^2 – 46^2 = 218^2 – 134^2 = 29568

z = 202 … y = 178 … x = 167 … c = 137
w = 134 … b = 94 … d = 119 … a = 71
202^2 – 178^2 = 167^2 – 137^2 = 134^2 – 94^2 = 119^2 – 71^2 = 9120
137^2 – 71^2 = 178^2 – 134^2 = 13728

z = 250 … y = 230 … x = 212 … c = 188
w = 166 … b = 134 … d = 140 … a = 100
250^2 – 230^2 = 212^2 – 188^2 = 166^2 – 134^2 = 140^2 – 100^2 = 9600
188^2 – 100^2 = 230^2 – 166^2 = 25344

Paul.

• benvitalis says:

I posted another solution with z = 334