Difference between revisions of "2023 IOQM/Problem 4"
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− | <math>(x^{2}+ 1)(x-1)(x+1) =(x-1)(y^{3}-23)-2</math>. Now [[divide]] the [[equation]] by <math>x-1</math> to get <cmath>(x^{2}+ 1)(x+1) = (y^{3}-23)-\frac{2}{x-1}</cmath> Since <math>x</math> and <math>y</math> are [[integers]], this implies <math>x-1</math> [[divides]] 2, so possible values <math>x-1</math> are -1, -2, 1, 2 | + | <math>(x^{2}+ 1)(x-1)(x+1) =(x-1)(y^{3}-23)-2</math>. Now [[divide]] the [[equation]] by <math>x-1</math>(considering that <math>x\neq 1</math>) to get <cmath>(x^{2}+ 1)(x+1) = (y^{3}-23)-\frac{2}{x-1}</cmath> Since <math>x</math> and <math>y</math> are [[integers]], this implies <math>x-1</math> [[divides]] 2, so possible values <math>x-1</math> are -1, -2, 1, 2 |
This means <math>x</math>= 0, -1(Rejected as <math>x</math> is a [[positive integer]]), 2, 3. so <math>x</math> =2 or 3. Now checking for each value, we find that when <math>x</math>=2, there is no [[integral]] value of <math>y</math>. When <math>x</math>= 3, <math>y</math> evaluates to 4 which is the only possible [[integral]] solution. | This means <math>x</math>= 0, -1(Rejected as <math>x</math> is a [[positive integer]]), 2, 3. so <math>x</math> =2 or 3. Now checking for each value, we find that when <math>x</math>=2, there is no [[integral]] value of <math>y</math>. When <math>x</math>= 3, <math>y</math> evaluates to 4 which is the only possible [[integral]] solution. |
Revision as of 10:23, 25 October 2023
Problem
Let be positive integers such that
Find the maximum possible value of .
Solution1(Diophantine)
, subtracting 1 on both sides we get
factorizing the LHS we get
. Now divide the equation by
(considering that
) to get
Since
and
are integers, this implies
divides 2, so possible values
are -1, -2, 1, 2
This means = 0, -1(Rejected as
is a positive integer), 2, 3. so
=2 or 3. Now checking for each value, we find that when
=2, there is no integral value of
. When
= 3,
evaluates to 4 which is the only possible integral solution.
So, = 3+ 4 =
~SANSGANKRSNGUPTA(inspired by PJ AND AM sir)