Number Theory (Thailand Math POSN 1st elimination round 2014)

Write a full solution.

1.) Let aa be an even number and bb an odd number such that (a,b)=1(a,b) = 1. Find the value of (54a1,52b1)\left(5^{4a}-1,5^{2b}-1\right) using Euclidean algorithm.

2.) (same as last year) Prove that if pp and 8p2+18p^{2}+1 are prime numbers, then 8p2+2p+18p^{2}+2p+1 is also prime number.

3.) Find all positive integers nn such that

12557+22557++n2557+(n+1)25571^{2557} + 2^{2557} + \dots + n^{2557} + (n+1)^{2557}

is a composite number.

4.) Let nNn \in \mathbb{N}. Prove that 22n+1+22n+12^{2^{n+1}} + 2^{2^{n}} + 1 has at least n+1n+1 distinct prime factors.

5.) Let a,b,cNa,b,c \in \mathbb{N} such that (a,b,c)=1(a,b,c) = 1 and 1a+1b=1c\displaystyle \frac{1}{a}+\frac{1}{b} = \frac{1}{c}. Prove that a+ba+b is a perfect square.

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Thailand Math POSN 2013

Thailand Math POSN 2014

Note by Samuraiwarm Tsunayoshi
6 years, 9 months ago

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1) is obviously 24.2) obviously p=3, which gives a satisfying result(79 is prime).3) is always composite (check divisibility by n or n+1) 5) use SFFT to get (a-c)(b-c)=c^2.But since they are coprime, we get the only sol c^2+c+1,c+1, whose sum is (c+1)^2

Bogdan Simeonov - 6 years, 9 months ago

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Q.5 Is from RMO-1992(Q.2)

Aneesh Kundu - 6 years, 9 months ago

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1)choosing smallest no. of b ( b=1) ...... 52b5^{2b}-1 = 525^{2}-1 = 24

similarly choosing smallest no. of a( a=2) ..... 54a5^{4a}-1 = 585^{8}- 1 (divides 24 as follows:)

521(mod24)5^{2} \equiv 1 \pmod{24}

=> 581(mod24)5^{8} \equiv 1 \pmod{24}

So gcd = 24

3) when no. divided by 2 leaves remainder 0 . no. is composite.

11(mod2)1 \equiv 1 \pmod{2}

115571(mod2)1^{1557} \equiv 1 \pmod{2}...... (1)

31(mod2)3 \equiv -1 \pmod{2}

315571(mod2)3^{1557} \equiv -1 \pmod{2}.......(2)

Adding (1) & (2) ....

11557+315570(mod2)1^{1557}+ 3^{1557} \equiv 0 \pmod{2}

all even to some power will be composite .

So , n= 3,7, 11 ..... should be the nos.

Ayush Choubey - 6 years, 8 months ago

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Someone told me that no.3) every positive integers make the sum composite, but I don't know how to prove that.

Samuraiwarm Tsunayoshi - 6 years, 8 months ago

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