![]() There is also one for fifth powers, the only one so far, given in Waldby’s database as: = There are solns to a 4+2b 4+2c 4 = d 4, the smallest of which is the Pythagorean-like: =. Which explains why Ramanujan’s constant (and others) exceeds a cube by nearly this amount, e π√163 ≈ 640320 3 + 744, though this property of 744 5 is probably unrelated.) ( Note: Incidentally, the number 744 figures prominently in the j-function j(τ) since this is the constant term of its series expansion, The first fifth power expressible as the sum of five positive fifth powers in two ways was found by Waldby as 744, The smallest number whose third power is expressible as the sum of three positive third powers in two ways is 41 as,įor fourth powers as four fourth powers in two ways, Jarek Wroblewski found it to be 31127 (see Wroblewski's database here ), (For seven 5 th powers equal to unity, there are formulas, also dependent on Pell eqns, to be given later.)Ĭ) x 1 5+ x 2 5+ x 3 5+ x 4 5+ x 5 5 = y 1 5+ y 2 5+ y 3 5+ y 4 5+ y 5 5 = z 5 No soln is yet known for 1+ x 2 5+ x 3 5+ x 4 5+x 5 5= z 5 with positive x i, as seen in James Waldby's database with z < 10,000. When all terms are integers with one equal to unity, it is unknown if a parametrization can exist, unlike for third powers where there is an infinite family which recent work by this author has shown involves a Pell equation. Tomita gave a complete table of = for positive x i < 1000, and Moore made a bigger table with sum almost 18,000 5. ![]() The smallest was by Lander, Parkin, and Selfridge, while three others were later given by Seiji Tomita, which were independently found by Duncan Moore who also gave four more: Just like three 3rd powers of signed integers can sum to 1 (a famous example of which involves the taxicab number 1729 = 1 3 + 12 3 = 9 3 + 10 3), five 5th powers can also do so. Whether this is reducible to solving an elliptic curve like for the fourth power version x 1 4+ x 2 4+ x 3 4 = x 4 4 still remains to be seen. ![]() This is a counter-example to Euler’s sum of powers conjecture and, for fifth powers, there are only three known so far:
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