MathDB

3

Part of 2010 Dutch IMO TST

Problems(2)

n|(p-1) and p | (n^3-1) => (4p-3) square

Source: 2010 Dutch IMO TST1 P3

1/10/2020
Let n\ge  2 be a positive integer and pp a prime such that n∣p−1n|p-1 and p∣n3−1p | n^3-1. Show 4p−3 4p-3 is a square.
number theoryPerfect Square
M(a, b) = a -1/b +b(b + 3/a) is integer => square also

Source: 2010 Dutch IMO TST2 p3

1/10/2020
(a) Let aa and bb be positive integers such that M(a,b)=a−1b+b(b+3a)M(a, b) = a - \frac1b +b(b + \frac3a) is an integer. Prove that M(a,b)M(a,b) is a square. (b) Find nonzero integers aa and bb such that M(a,b)M(a,b) is a positive integer, but not a square.
number theoryPerfect Square