Answer: Another Putnam Problem (Dec 2, 2007)
Saturday, 8 of December , 2007 at 12:57 pm
Apparently, I bit off more than I could chew when tackling this question. I don’t have formal mathematical solution to this problem, but I will show you what I came up with.
Question: Determine the maximum value of 
, where m and n are integers in the range 1,2,…,1981 satisfying 
.
Analysis: In an attempt to maximize , we should first investigate what values of m and n satisfy the constraints which can be reduced to 
. If we look for solutions to this equation for small values of m and n, you will find the following sequence:
The solution appears to be Fibonacci numbers. This would mean that if (m, n) are solutions then (n, m+n) would also be a solution. If we put (n, m+n) into the constraints, we can verify that all Fibonacci numbers are a solution.
This shows that the following sequence is a solution to the given constraints:
The part that I wasn’t able to prove is that the Fibonacci numbers are the ONLY solution to the constraints. However, if they were, the function would be maximized when 
.
In order to prove that this was the correct solution, I wrote the following perl script:
$max_val = 0;
$max_m = 0;
$max_n = 0;
for ($m = 1; $m < 1982; $m++) {
for ($n = 1; $n < 1982; $n++) {
$val = $n*$n - $m*$n - $m*$m;
$val = $val*$val;
if($val == 1){
$val = $m*$m + $n*$n;
if($val > $max_val){
$max_val = $val;
$max_m = $m;
$max_n = $n;
}
}
}
}
print "Max Value: ".$max_val." m: ". $max_m." n: ".$max_n;
This script confirmed that the maximum value is when
$ perl find.pl
Max Value: 3524578 m: 987 n: 1597
Apparently, I bit off more than I could chew when tackling this question. I don’t have formal mathematical solution to this problem, but I will show you what I came up with.
Question: Determine the maximum value of , where m and n are integers in the range 1,2,…,1981 satisfying .
Analysis: In an attempt to maximize , we should first investigate what values of m and n satisfy the constraints which can be reduced to . If we look for solutions to this equation for small values of m and n, you will find the following sequence:
The solution appears to be Fibonacci numbers. This would mean that if (m, n) are solutions then (n, m+n) would also be a solution. If we put (n, m+n) into the constraints, we can verify that all Fibonacci numbers are a solution.
This shows that the following sequence is a solution to the given constraints:
The part that I wasn’t able to prove is that the Fibonacci numbers are the ONLY solution to the constraints. However, if they were, the function would be maximized when .
In order to prove that this was the correct solution, I wrote the following perl script:
$max_val = 0; $max_m = 0; $max_n = 0; for ($m = 1; $m < 1982; $m++) { for ($n = 1; $n < 1982; $n++) { $val = $n*$n - $m*$n - $m*$m; $val = $val*$val; if($val == 1){ $val = $m*$m + $n*$n; if($val > $max_val){ $max_val = $val; $max_m = $m; $max_n = $n; } } } } print "Max Value: ".$max_val." m: ". $max_m." n: ".$max_n;
This script confirmed that the maximum value is when
$ perl find.pl Max Value: 3524578 m: 987 n: 1597
Category: Answers, Putnam Problems
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