Determine all triplets (a,b,c) of nonzero integers satisfying:
987654321*a + 123456789*b + c = (a + b + c)³

Prove that there are no others.

Let x = a + b + c

987654321*x > x^3
987654321 > x^2
x < 31427

x^3 > 987654321
x > 995

DEFDBL A-Z
FOR t = 995 TO 31427
    IF t MOD 1000 = 0 THEN PRINT t;
    FOR a = 1 TO t - 2
        FOR b = 1 TO t - a - 1
            c = t - a - b
            lhs = 987654321 * a + 123456789 * b + c
            sum = a + b + c
            rhs = sum * sum * sum
            IF lhs = rhs THEN
                PRINT a; b; c, lhs
            END IF
        NEXT
    NEXT a
NEXT t

has found so far

 a      b      c
450   4500   5050  results in 1000000000000 for each side.

It's still chugging away and has checked beyond a+b+c = 12000 so far, but not yet up to 13000.

Posted by Charlie on 2013-06-09 15:01:35