He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729.
It was proposed by Leonhard Euler in 1769.
Norrie, 1911) This is the smallest solution to the problem by R.
Norrie. ====== : (Lander & Parkin, 1966) : (Lander, Parkin, Selfridge, smallest, 1967) : (Sastry, 1934, third smallest) ====== : (M.
Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for .
A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: : (Lander & Parkin, 1966), : (Scher & Seidl, 1996), and : (Frye, 2004). In 1988, Noam Elkies published a method to construct an infinite series of counterexamples for the case.
Norrie. ====== : (Lander & Parkin, 1966) : (Lander, Parkin, Selfridge, smallest, 1967) : (Sastry, 1934, third smallest) ====== : (M.
This solution is the only one with values of the variables below 1,000,000. == Generalizations == In 1967, L.
Norrie. ====== : (Lander & Parkin, 1966) : (Lander, Parkin, Selfridge, smallest, 1967) : (Sastry, 1934, third smallest) ====== : (M.
A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: : (Lander & Parkin, 1966), : (Scher & Seidl, 1996), and : (Frye, 2004). In 1988, Noam Elkies published a method to construct an infinite series of counterexamples for the case.
Substituting into the identity and removing common factors gives the numerical example cited above. In 1988, Roger Frye found the smallest possible counterexample : for by a direct computer search using techniques suggested by Elkies.
A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: : (Lander & Parkin, 1966), : (Scher & Seidl, 1996), and : (Frye, 2004). In 1988, Noam Elkies published a method to construct an infinite series of counterexamples for the case.
Dodrill, 1999) ====== : (S.
Chase, 2000) ==See also== Jacobi–Madden equation Prouhet–Tarry–Escott problem Beal's conjecture Pythagorean quadruple Generalized taxicab number Sums of powers, a list of related conjectures and theorems == References == == External links == Tito Piezas III, A Collection of Algebraic Identities Jaroslaw Wroblewski, Equal Sums of Like Powers Ed Pegg Jr., Math Games, Power Sums James Waldby, A Table of Fifth Powers equal to a Fifth Power (2009) R.
As of 2002, there are no solutions for k=6 whose final term is ≤ 730000. ====== : (Plato's number 216) This is the case a = 1, b = 0 of Srinivasa Ramanujan's formula :(3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3 .
A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: : (Lander & Parkin, 1966), : (Scher & Seidl, 1996), and : (Frye, 2004). In 1988, Noam Elkies published a method to construct an infinite series of counterexamples for the case.
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