Problems need HardWork 12
Number Theory
Level
5
Find the number of integer pairs of which satisfy above equation.
The answer is 6.
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1 solution
One can write the original Diophantine equation as:
x 4 + x 3 + x 2 + x = y 2 − 1 ;
or x ( x + 1 ) ( x 2 + 1 ) = ( y + 1 ) ( y − 1 ) .
Clearly, the integral ordered-pairs ( 0 , 1 ) ; ( 0 , − 1 ) ; ( − 1 , 1 ) ; ( − 1 , − 1 ) are solutions. Now we need to check the following systems:
x 2 + 1 = y − 1 & x 2 + x = y + 1 (i)
x 3 + x = y + 1 & x + 1 = y − 1 (ii)
System (i) yields ( 3 , 1 1 ) as a solution, as well as ( 3 , − 1 1 ) due to squaring y in the original Diophantine equation. System (ii) yields ( 3 1 / 3 , 3 1 / 3 + 2 ) , which is an irrational (non-integral) pair.
Hence there are six integral pairs:
( 0 , 1 ) ; ( 0 , − 1 ) ; ( − 1 , 1 ) ; ( − 1 , − 1 ) ; ( 3 , 1 1 ) ; ( 3 , − 1 1 ) .