Euler's conjecture (1769) states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a k-th power, then n is greater than or equal to k.

For k=2 (3,4,5) and for k=3 (3,4,5,6) support this conjecture,but it was later proven false for k=4 and 5.
To find 5 integers such that a^4+b^4+c^4=d^4 is extremely difficult (big,big numbers), but for fellows floobers to solve a^5+b^5+c^5+d^5=e^5 is a piece of cake, since (HINT) all the integers are below 200(sic!).
OK, it is prestigious, albeit easy task:
Defeat Euler for k=5 .

DEFDBL A-Z
FOR a = 1 TO 199
    PRINT a;
    t1 = a * a * a * a * a
    FOR b = a TO 199
        t2 = b * b * b * b * b
        FOR c = b TO 199
            t3 = c * c * c * c * c
            FOR d = c TO 199
                t4 = d * d * d * d * d
                sum = t1 + t2 + t3 + t4
                root5 = INT(sum ^ (1 / 5) + .5)
                IF root5 * root5 * root5 * root5 * root5 = sum THEN
                    PRINT
                    PRINT a, b, c, d, root5
                END IF

            NEXT
        NEXT
    NEXT
NEXT

 

finds (a,b,c,d) = (27,84,110,133) resulting in 144^5.

Posted by Charlie on 2013-06-25 12:55:38