On a class of diophantine equations
Cohn (1971) has shown that the only solution in positive integers of the equation Y(Y+1)(Y+2)(Y+3)=2X(X+1)(X+2)(X+3) is X=4, Y=5. Using this result, Jeyaratnam (1975) has shown that the equation Y(Y+m)(Y+2m)(Y+3m)=2X(X+m)(X+2m)(X+3m) has only four pairs of nontrivial solutions in integers given by...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007858 |
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| Summary: | Cohn (1971) has shown that the only solution in positive
integers of the equation Y(Y+1)(Y+2)(Y+3)=2X(X+1)(X+2)(X+3) is X=4, Y=5. Using this result, Jeyaratnam (1975) has shown that the
equation Y(Y+m)(Y+2m)(Y+3m)=2X(X+m)(X+2m)(X+3m) has only four
pairs of nontrivial solutions in integers given by X=4m or
−7m, Y=5m or −8m provided that m is of a specified type.
In this paper, we show that if m=(m1,m2) has a specific form
then the nontrivial solutions of the equation Y(Y+m1)(Y+m2)(Y+m1+m2)=2X(X+m1)(X+m2)(X+m1+m2) are m times the primitive solutions of a similar equation with smaller m's. Then we specifically find all solutions in integers of the
equation in the special case m2=3m1. |
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| ISSN: | 0161-1712 1687-0425 |