# Tagged Questions

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### Diophantine Equations problem 2

Find all the solutions to the Diophantine equation x^2+y^2=2(z^2) .I do not have alot of expirience on Diophantine equations and i do not know how to approximate them.I can see that the tripples of ...
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### Diophantine equation exercise [duplicate]

Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not ...
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### Power of prime number in $n!$ - prove this formula

How to prove: for $l := \max\{i \in \mathbb{N}_0 : p^i \mid n!\}$ it holds: a) $l = \sum_{i=1}^\infty [\frac{n}{p^i}]$ b) $l \leq [ \frac{n}{p-1} ]$ ? If I take a closer look at the sum in a) I ...
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### Find all integers n such that n−2014 and n+ 2014 are both triangular numbers.

I came across this problem when searching for triangular numbers questions. I know that I need to use the equation, $$\frac {n(n+1)}{2}$$ but I don't know how to apply it to this problem.
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### GCD and LCM Problem

Let $x$ and $y$ be positive integers, $x < y$, and $x + y = 667$. Find all pairs $(x,y)$ if $\text{lcm}\,(x,y)/\gcd\,(x,y) = 120$. This problem was from my number theory homework, and I don't get ...
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### Prove that if $\gcd(a, n)=1$, then $n\mid a^k-1$ for some $k$

How can I show that if $a$ and $n$ are natural numbers with the condition that $\gcd(a,n)=1$, then there exists a natural number $k$ such that $n \mid a^{k}-1$ What I tried doing is set it up in mod ...
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### About the calculation of decimal digits of series up to the nth digit

Considering that we don't know any of the digits of some number defined as the limit up to infinity of a sum, I want to know how many terms do I have to sum to get the correct decimal representation, ...
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### prove of $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$ [duplicate]

i came across this answer and i saw the given solution but i can not understand how it proves the given problem. Ok i get that $lcm(5,7)= 35$ and it is the same as the $(mod 35)$. Please can someone ...
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### Modular equations, find x

Problem: Find an integer $x$ such that $x = 5\pmod 8, x = 3 \pmod 9, x = 4 \pmod 7$. Attempt: By the Chinese Remainder Theorem " Suppose $a_1,a_2,...a_k$ are integers pairwise relatively prime ...
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### What is a non Trivial Square Root?

I need to understand the concept behind a non trivial square root. Also how to answer these two questions and how to get to the answer? Give a non-trivial square root of 30 Give a non-trivial ...
### How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$?
How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$ ? I'm new to elementary number theory and I'm not sure what to do AT ALL. We're currently studying primitive roots and indices.