1234525286
Number Theory
Level
3
How many ordered pairs of positive integers satisfy ?
The answer is 0.
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1 solution
Utilizando o critério da divisibilidade de um número por 4 conclui-se que 1234525286 deixa resto 2 na divisão por 4,então basta analisarmos os restos de x^2-y^2 na divisão por 4. Observe que x^2 só pode assim como y^2, deixar resto 0 ou 1 na divisão por 4,então:
x^2-y^2 só pode deixar 0,1 ou 3 como resto na divisão por 4 oque impossibilita que haja solução para a equação do problema.
Rough translation :
Using the divisibility of a number of criteria for 4 concluded that 1234525286 2 leaves a remainder when divided by 4, then simply analyzing the remainder of x 2 − y 2 when divided by 4. Note that x 2 only as y 2 , let rest 0 or 1 on division by 4, then:
x 2 − y 2 can only leave 0.1 or 3 as a remainder in the division by 4. So it is impossible that there is solution to the equation of the problem.