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On the Four Integer Cubes Problem
Author(s) -
Mordell L. J.
Publication year - 1957
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-32.3.383-t
Subject(s) - integer (computer science) , combinatorics , sierpinski triangle , infinity , square (algebra) , mathematics , discrete mathematics , computer science , geometry , fractal , mathematical analysis , programming language
In my paper “ On the four integer cubes problem ”, Journal London Math. Soc , 11 (1936), 208–218,1 prove the result that if a, b, c, d are integers, there are an infinity of integer solutions of x 3 + y 3 + z 3 + w 3 = a 3 + b 3 + c 3 + d 3 when any one of the three numbers −( a + b )( c + d ), etc. is positive and not a perfect square. Prof. Sierpinski has kindly pointed out that the proof does not hold when a = b, c = d , etc. Hence it is not known if x 3 + y 3 + z 3 + w 3 = 2 a 3 +2 b 3 has an infinity of integer solutions. In fact, the result stated at the end of page 216, that an infinity of solutions exist, is obviously subject to the further condition AC 2 + BD 2 ≠ 0, i.e. (a + b )( a − b ) 2 + ( c + d )( c − d ) 2 ≠ 0. Since − AB is not a perfect square, this inequality is satisfied unless C = 0, D = 0, i.e. unless a = b, c = d .