Equation : (x – y)^n = x*y

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

equation-x-y-n-1

 
 

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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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