Symmetric Does Not Imply Equal

Number Theory Level 5

Find the number of pairs of non-negative integers ( n , m ) (n,m) , such that 1 ≤ n < m ≤ 100 1\leq n< m\leq 100 , n ∣ m 2 − 1 n \mid m^2-1 and m ∣ n 2 − 1 m \mid n^2-1 .


The answer is 208.

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Calvin Lin by Calvin Lin

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2 solutions

Calvin Lin Staff
Mar 31, 2014

Hint: Vieta Root Jumping

Many people who spotted the patterns of ( n , n + 1 ) (n, n+1) , ( 1 , n ) (1, n) and ( n , n 2 − 1 ) (n, n^2 - 1) as possible solutions, missed out the solution sets of ( 8 , 21 ) , ( 21 , 55 ) , ( 15 , 56 ) (8,21), (21, 55), (15, 56) .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

It might be a typo but (12,56) is not a solution. :)

Jan Frayre - 7 years ago

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Thanks. I've updated it to ( 15 , 56 ) (15, 56) , which arose from the pair ( 4 , 15 ) (4, 15) .

Calvin Lin Staff - 6 years, 8 months ago
Scott Kominers
Apr 11, 2014

This problem can be solved with the following Mathematica command: 

Total[Total[Array[If[Divisible[(#2)^2 - 1, #1] && Divisible[(#1)^2 - 1, #2] && #1 < #2, 1, 0] &, {100, 100}]]]
1 Helpful 0 Interesting 0 Brilliant 0 Confused

Absolutely beautiful. :D

Finn Hulse - 7 years, 1 month ago

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