Difference between revisions of "2023 IOQM/Problem 4"

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(Problem)
 
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<cmath>x^{4}=(x-1)(y^{3}-23)-1</cmath>
 
<cmath>x^{4}=(x-1)(y^{3}-23)-1</cmath>
  
Find the maximum possible value of <math>x + y</math>.
+
Find the maximum possible value of <math>x + y</math>
x⁴=(x-1)(y³-23)-1
 
x⁴-1=(x-1)(y³-23)-2
 
(x²-1)(x²+1)=(x-1)(y³-23)-2
 
(x-1)(x+1)(x²+1)=(x-1)(y³-23)-2
 
(x+1)(x²+1)=(y³-23)-(2⁄x-1)
 
x≠1, x is an integer so x-1|2
 
thus x-1≼2, x≼3, thus x= 2 or 3
 
For x=2 , (2+1)(4+1)=(y³-23)-(2/2-1)
 
(3)(5)=(y³-23)-2
 
15=(y³-23)-2
 
y³= 15+2+23
 
y³=40, but y is an integer and 40 is not an perfect cube
 
thus x≠2
 
For x=3 , (3+1)(9+1)=(y³-23) - (2/3-1)
 
(4)(10)=(y³-23)-1
 
40+1=y³-23
 
y³=41+23
 
y³=64, y=4
 
thus , x=3,y=4 , so x+y= 3+4=7
 
So the answer of this question will be 7
 

Latest revision as of 23:30, 31 August 2024

Problem

Let $x, y$ be positive integers such that \[x^{4}=(x-1)(y^{3}-23)-1\]

Find the maximum possible value of $x + y$