## Equation : (x – y)^n = x*y

Find triplets   (x, y, n)   of positive integers such that

Note that if we set

$x \; = \; (k+1) \, (k^2+k)$
$y \; = \; k \, (k^2+k)$

$x \; - \; y \; = \; (k+1) \, (k^2+k) \; - \; k \, (k^2+k) \; = \; k \, (k + 1)$

$x \, y$
$= \; ((k+1) \, (k^2+k)) \, (k \, (k^2+k))$
$= \; k \, (k + 1) \, (k^2 + k)^2$
$= \; k^3 \, (k + 1)^3$

Hence   $n \; = \; 3$

and,

$x \, / \, y \; = \; (k + 1) \,/ \,k$

math grad - Interest: Number theory
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### 3 Responses to Equation : (x – y)^n = x*y

1. Paul says:

Here’s a few

(4 – 2)^3 = 4 * 2
(18 – 12)^3 = 18 * 12
(48 – 36)^3 = 48 * 36
(100 – 80)^3 = 100 * 80
(180 – 150)^3 = 180 * 150
(294 – 252)^3 = 294 * 252
(448 – 392)^3 = 448 * 392
(648 – 576)^3 = 648 * 576
(900 – 810)^3 = 900 * 810

Paul.

2. Paul says:

and some more

(1210 – 1100)^3 = 1210 * 1100
(1584 – 1452)^3 = 1584 * 1452
(2028 – 1872)^3 = 2028 * 1872
(2548 – 2366)^3 = 2548 * 2366
(3150 – 2940)^3 = 3150 * 2940
(3840 – 3600)^3 = 3840 * 3600
(4624 – 4352)^3 = 4624 * 4352
(5508 – 5202)^3 = 5508 * 5202
(6498 – 6156)^3 = 6498 * 6156
(7600 – 7220)^3 = 7600 * 7220
(8820 – 8400)^3 = 8820 * 8400
(10164 – 9702)^3 = 10164 * 9702
(11638 – 11132)^3 = 11638 * 11132
(13248 – 12696)^3 = 13248 * 12696
(15000 – 14400)^3 = 15000 * 14400
(16900 – 16250)^3 = 16900 * 16250
(18954 – 18252)^3 = 18954 * 18252
(21168 – 20412)^3 = 21168 * 20412
(23548 – 22736)^3 = 23548 * 22736
(26100 – 25230)^3 = 26100 * 25230
(28830 – 27900)^3 = 28830 * 27900
(31744 – 30752)^3 = 31744 * 30752
(34848 – 33792)^3 = 34848 * 33792
(38148 – 37026)^3 = 38148 * 37026
(41650 – 40460)^3 = 41650 * 40460
(45360 – 44100)^3 = 45360 * 44100
(49284 – 47952)^3 = 49284 * 47952
(53428 – 52022)^3 = 53428 * 52022
(57798 – 56316)^3 = 57798 * 56316
(62400 – 60840)^3 = 62400 * 60840
(67240 – 65600)^3 = 67240 * 65600
(72324 – 70602)^3 = 72324 * 70602
(77658 – 75852)^3 = 77658 * 75852
(83248 – 81356)^3 = 83248 * 81356
(89100 – 87120)^3 = 89100 * 87120
(95220 – 93150)^3 = 95220 * 93150

Paul.

• benvitalis says:

Yes. It seems only to work for n = 3. Can you find a counterexample?