# Tagged Questions

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### A conjecture on partitions

While trying to prove a result in group theory I came up with the following conjecture on partitions: Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for ...
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### Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$

Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is ...
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### Function producing primes continuously

I believe that there is no polynomial function gives primes continuously. This statement says that, we can create some other function (other than polynomial) and can find primes. I did my trail and ...
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### simplify this complex expression [on hold]

Simplify this expression : I want to evaluate this expression modulo 10^9+7 $$\frac{\binom{N/2}{b-1}}{\binom{N}{b-1}}\cdot \frac{N}{N-(b-1)}\cdot N!$$ where N=$2^k$ for some given k<=20. How to ...
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### remainder of the division of $7^{1203}$ by $143$

I have to find the remainder of the division of $7^{1203}$ by $143$. I thought that I could use the Euler Theorem: Let $a \in \mathbb{Z}$ and $n \in \mathbb{N}$.We also know that $(a,n)=1$.Then ...
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### Primality of $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , when $q$ is prime, $j\ge0$?

Let $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , $q$ prime and $j\ge0$. $P_{2,j}$ is a Fermat number, $P_{q,0}$ is a Mersenne number. Apart from Fermat primes and Mersenne primes, and apart from ...
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### Prove that if a,b & k are natural numbers such that $k^{\frac{a}{b}}$ is rational, then $k^{\frac{a}{b}}$ is a natural number.

Proposed solution: we know that $k^{\frac{a}{b}}=\frac{m}{n}$is rational,where m,n $\in$ Z. Assume w.l.o.g. that (m,n)=1. Then $nk^{\frac{b}{a}}=m$ $n^{a}k^{b}=m^{a}$ Then we write $n^a,k^b,m^a$ in ...
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### Square freeness of Fermat numbers

I will look into the squarefreeness of the Fermat numbers once again. This is also an open problem in Number Theory. My primary source of knowledge comes from the book Fermat Numbers From Number ...
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### Linear Diophantine Equations

I was asked to find i) all integer solutions, and ii) all non-negative integer solutions to the equations below. I know (a) has no answers, but have no idea how to go about proving the rest. ...
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### Help with proving bezout's theorem?

I cannot figure out how to prove this Theorem!! Any help would be appreciated.
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### how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
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### Primes and infinite primes of the form

can you give the validity or proof of the following statements of my observations on Primes? $(1)$ For a positive integer $k$, there exists infinitely many primes of the form $29 + 72k$. $(2)$ If the ...
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### $(a,b) = 1$ implies $a|n$ and $b|n \implies ab|n$

Prove if $(a,b) = 1$ implies $a|n$ and $b|n \implies ab|n$. I'm pretty sure this has been asked before but I cannot find anything online.... I also have no idea how to solve it, I get stuck with al = ...
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### Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator? In order to say clearly, this number should given by a certain formula, such as ...
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### A club for some special prime numbers: new members welcome

Given an integer $i$, find an integer $n$ ( $2^{j-1}\le n <2^j$), and a prime divisor $p$ of $M_n=2^n-1$, so that $v= j+i$; where $p$ is written as $k2^v+1$, $k$ odd. In other words, $j$ is such ...
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### Representation as a sum of two squares

Recently, I have been wandering that what natural numbers can can be written as a sum of squares of two coprime positive integers in two different ways, order being irrelevant, or with the help of ...
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### Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ ...
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### Is $\sum_{j=1}^{k}a^j \equiv 0 \mod p\;$? where k is the order of $a \mod p$, with $p$ being an odd prime?

In other words is $a^1 + a^2 + \dotsm a^k \equiv 0\mod p\;$? This is true when $a$ is a primitive root of $p$ because $a^1, a^2, \dotsc a^k$ are congruent to $1,2,\dotsc,p-1$ in some order. Hence, ...
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### Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
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### Twelve Distinct Positive Integers

Let S be a set of twelve distinct positive integers such that for distinct a, b, c, and d in S, a + b ≠ c + d. Show that the largest element in S is greater than 56. I found some math competition ...
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### If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$\text{If m,n\in \mathbb{Z}_+ such that 3m^2+m=4n^2+n, then (m-n) is a perfect square.}$$ I have managed to partially prove ...
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### Last Digit Of N^M

Given $N,M$ What is the best way to find last digit of $N^M$ if both $N,M$ Can be as large as $10^{18}$? EXAMPLE : if $N=2$ and $M=4$ then answer would be $6$.
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### Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
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### Find the last two digits of $3^{45}$

I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or ...
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### Prove a property of the divisor function

Let $q$ be an odd composite integer and $\sigma(q)$ the sum of the positive divisors of $q$. For what $q$ is it true that $$(\sigma(q)-q) \mid (q-1) \;?$$ If $q$ is prime, it is clear that it is ...
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### Probability that a number is a generator of a subgroup of $Z_p^*$ of order $q$

If $p$ is a random, large prime, what is the probability that a random element $x$ in $Z_p$ is a generator of a subgroup of $Z_p^*$ of order $q$, where $q$ is the largest number in the prime ...