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Solution of a diophantine problem. (English) Zbl 0544.10011

The author proves that the only positive integers which are simultaneously triangular (i.e., of the form \(y(y+1)/2)\) and pyramidal (i.e., of the form \(x(x+1)(2x+1)/6)\) are \(1, 55, 91\) and \(208\,335\). The result is obtained by determining all integer points on the elliptic curve \(3y^ 2=x^ 3-x+3\). The right hand side is decomposed in a suitable cubic field, and considering the residues of \(x \bmod 3\) reduces the problem to solving eight equations, a typical example being \(4\,U^ 4- 36\,UV^ 3+9\,V^ 4=1\). Then these eight equations are solved in integers using the arithmetic of quartic fields. The proofs are complete and fairly detailed.

MSC:

11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
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