# Questions tagged [elementary-number-theory]

For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.

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### $x^5=y^2+10$ has no solutions

I am looking for an elementary way to show the equation $x^5=y^2+10$ has no integer solutions. I have checked the equation mod $n$ for $n<1000$ and it had solutions every time. Here is my proof, ...
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### Manual Primality Testing methods

I am curious to know some other interesting manual methods for Primality testing. Here is one of the methods I know as of now. Suppose let us say, we need to check whether $397$ is Prime or not. We ...
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### Using FTA to prove exponential relation between integers [duplicate]

Question: Show that if $x$ and $y$ are non-zero integers and $x^2 = y^3$ then $x=a^3$ and $y=b^2$ for some integers $a$ and $b$ My attempt: If $y^3 = x^2$ then this means $y$ is a natural number, ...
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### Are there sets that are so unique that even multisets can't share a sum with them?

For sets that don't have $1$s or $0$s and all numbers are whole numbers. We already know $A$ = {$a,b,c...$} and it has distinct subset sums. For multisets consisting of only elements from $A$, can we ...
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### How to determine if the following expression is necessarily an integer [duplicate]

How can I prove that: $$\frac{30!}{(10!)^3} \in \mathbb Z$$ The question spontaneously occurred to me, but I'm unsure about the best approach to answering it. I apologize for the brevity of my ...
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### If $\varphi(n)=2p$ then $p$ is a Sophie Germain Prime [closed]

Define $n,p\in\mathbb{N}$ with p prime. I'm struggling to show that if $\varphi(n)=2p$, then $2p+1$ is prime.
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### How can I prove this statement about a Diophantine relation?

I am after a proof of the following Diophantine relation. Below I have the proof constructed so far. This is not for an assignment, just recreational math and adult learning. I would like to know if I'...
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### Squares of sum of digits of a number part 2

This is a sequel to my previous question which was this Squares of sum of digits of a number For those new I will give some context. Take a number then add up it's digits and square them then repeat ...
1 vote
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### Sequences of the form $A(n) = A(A(n-1)\bmod n)^2$

$$A(0)= x \in\mathbb{Z}^+,\ A(n) = A(A(n-1) \bmod n)^2$$ At first glance, one would think that such sequence would grow very fast. But my testing suggest that this sequence actually ends with $x^4$ ...
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1 vote
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### Consecutive numbers

Are there ever more consecutive composite numbers than there are primes up to that point? I imagine not, because the primes are the ones which cancel out multiples, so will inevitably have to fill in ...
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### Behaviour of $a^k\pmod b,\ k=1,2,3,\ldots$

Let $n\in\mathbb{N}.$ Suppose $\left(x_k\right)_{k=1}^{n}$ is a $k-$tuple of $-1$'s and $1$'s. Let (the function) $r(i,j)$ be the remainder when $i$ is divided by $j,$ so that $0\leq r(i,j) \leq j-1.$ ...
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1 vote
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### Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
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### A question on AOPS from AMC 12 left without a solution.

So I was doing some math questions on AOPS when i stumbled across this question which did not have a solution for it. I really want to know the solution to this problem so please help me. 2002 AMC 12P ...
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### Finding the 123ʳᵈ Number in a Sequence After Removing Multiples of 5 or 7

I'm working on a problem involving a sequence of the first 300 positive integers, from 1 to 300. However, I need to find the 123ʳᵈ number in the sequence after removing all numbers that are divisible ...
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### Is this a legitimate method of finding another prime number? [duplicate]

My motivation for asking this question stems from Euclid's elegantly simple proof of the infinitude of prime numbers. I am not suggesting an alternative proof, since my method, even if is valid, ...
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### GCD of $\frac{a^p + b^p}{a + b}$ and $a + b$ where p is an odd prime [duplicate]

The question is to find the GCD of $\frac{a^p+b^p}{a+b}$ and $a+b$ where p is an odd prime, in two cases, when $p|(a+b)$ and when it doesn't. I found, through several examples, that in the first case ...
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Primorial number system is a number sytem that uses primorials which are defined as follows : Let $p_1=2, p_2=3,p_3=5,p_4=7,p_5=11,...$ the primes. The sequence of primorials, noted $p_n\#$ is (p_n\#...