N=(abcabd) is a 6-digit number with d-c=1.

FIND N, given it is an even square.

Rem: (abcabd) denotes chaining (= concatenation), not a product

024025, 075076:

Let x+1000(x-1) =y^2: then find the integer solutions{n,x,y}: y^2, when the listing:
x = 1001n^2-2000n+1000,  y = 1000-1001n, n=1: {1,1,-1}:1
x = 1001n^2-1692n+716,   y = 846-1001 n,  n=0: {0 ,716, 846}: 715716
x = 1001n^2-1692n+716,   y = 846-1001 n,  n=1: {1,25,-155}: 024025
x = 1001n^2-1454n+529,   y = 727-1001 n,  n=0: {0,529,727}: 528529
x = 1001n^2-1454n+529,   y = 727-1001 n, n=1: {1,76,-274}: 075076
x = 1001n^2-1146n+329,   y = 573-1001 n, n=0: {0, 329, 573}: 328329
x = 1001n^2-1146n+329,   y = 573-1001 n, n=1: {1,184,-428}: 183184
x = 1001n^2-856n+184,   y = 428-1001 n,  n=0: {{0,184,428}: 183184
x = 1001n^2-856n+184,   y = 428-1001 n, n=1: {1,329,-573}: 328329
x = 1001n^2-548n+76,   y = 274-1001 n,  n=0: {0, 76, 274}: 075076
x = 1001n^2-548n+76,   y = 274-1001 n,  n=1: {1,529,-727}: 528529
x = 1001n^2-310n+25,   y = 155-1001 n,  n=0: {0,25,155}: 024025
x = 1001n^2-310n+25,   y = 155-1001 n,  n=1, {1,716,-846}: 715716
x = 1001n^2-2n+1,   y = 1-1001 n, n=0: {0,1,1} :1
x = 1001n^2-2n+1,   y = 1-1001 n, n=1, {1,1000,-1000}:  1000000

is exhaustive, I believe.

 

Edited on January 19, 2013, 8:10 am

Posted by broll on 2013-01-19 07:22:14