Questions tagged [number-theory]

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Are there techniques for quickly seeing patterns in numbers?

Recently I was watching 8 out of 10 cats does countdown (S18E01 - 26 July 2019), for which the math question there was the following numbers: if you haven't seen countdown, you are provided 6 numbers,...
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Find all solutions $(x, y, z)$ ∈ $\mathbb N^+ ×\mathbb N^+ ×\mathbb N^+$ for the following equations in $\mathbb Z$

Find all solutions $(x, y, z)$ ∈ $\mathbb N^+ ×\mathbb N^+ ×\mathbb N^+$ for the following equations in $\mathbb Z$: (a) $3x^2 + 4y^2 = z^2$ (b) $3x^2 + 6y^2 = z^2$ My attempt: a) If I ...
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Pairs of perfect squares

Two perfect squares are said to be friendly if one is obtained from the other by adding the digit 1 on the left. For example, $1225 = 35 ^ 2$ and $225 = 15 ^ 2$ are friendly. Prove that there are in ...
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Is there a perfect square (other than 9) all of whose digits are 7, 8, or 9?

Clearly, $3^2=9$ is a perfect square, all of whose digits are $7$, $8$, or $9$. Are there any other perfect squares with this property? This is an interesting question that does not seem to be ...
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p is without a common factor of 100000.Prove that power of p decimal representation ends with numbers group 00001. [on hold]

p is without a common factor of 100000.Prove that power of p decimal representation ends with numbers group 00001. Also prove that every natural nubmer n exist natural number k, that k power of p ...
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When is $\sum_{n=0}^{4}{x^n},~\text{where}~x\in \mathbb{N}>0$ square? [duplicate]

When is $\sum_{n=0}^{4}{x^n}$ ($x$ a natural number greater $0$) square ? Any help would be appreciated.
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Cyclotomic $\mathbb{Z}_p$-extensions of number fields

Let $p$ be a fixed odd prime. Let $F/\mathbb{Q}$ be any number field which does not contain $\mu_p$. Consider the cyclotomic $\mathbb{Z}_p$-extension of $F$, this is the unique $\mathbb{Z}_p$-...
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How many positive integer(s) x is / are there so that $\sqrt{x^2 + 4x + 670}$ is an integer? [on hold]

How many positive integer(s) x is / are there so that $\sqrt{x^2 + 4x + 670}$ is an integer? I can do it up till the completing the square portion. However, the (u+x+2)(u-x-2) messes me up.
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Asymptotics of the Atkin-Bernstein sieve on primes

I am working through the paper by Atkin and Bernstein, Prime sieves using binary quadratic forms, see https://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01501-1/S0025-5718-03-01501-1.pdf ...
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How to show that no function f from Z to Z satisfies f(f(x)) = x + 5? [duplicate]

Question pretty much explains it. So far I can show that f(x + 5) = f(x) + 5 but I don't know where to go from there.
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Check whether or not a given number N is triangular number [on hold]

I'm trying to write a program that would tell me wether or not a number N is triangular number. Any formula exists for this ?
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A Diophantine Equation related to arithmetic progression: $T_n=a^n+b^n+c^n$.

I'm working on a problem that for complex $a,b,c$, when will these four numbers $a+b+c, a^2+b^2+c^2, a^3+b^3+c^3, a^4+b^4+c^4$ becomes an arithmetic progression with integer values. If we let this ...
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Bound on the sum of squares of the number of divisors until $Z$ (Vaughan Circle method)

I was wondering how I could show that $\displaystyle \sum_{x\leq Z}d(x)^2\ll Z(\log 2Z)^3$, where $d(x)$ counts the number of divisors of $x$. I tried to prove this in the following fashion (and ...
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Prime factor $>250000$ for $1002004008016032$

I need to find a prime factor, $p$, of $1002004008016032$ such that $p \gt 250000$. Now this is a very large number and I know it would be stupid of me to factorize such a large number into its prime ...
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Jacobi Symbol: $\sum_{n=1}^{n=p}\left(\frac{n^2+a}{p}\right)=-1$

Show that if $(a,p)=1$, $p$ an odd prime then, $\sum_{n=1}^{n=p}\left(\frac{n^2+a}{p}\right)=-1$, where $\left(\frac{n^2+a}{p}\right)$ is the Jacobi symbol. This question has been taken from the book ...
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Calculating Legendre symbol : $\left(\frac{3}{2^{2^n}+1}\right)$ , n being positive.

I want to find the Legendre symbol : $\left(\frac{3}{2^{2^n}+1}\right)$ for any positive n. Also I want to find the Legendre symbol for: $\left(\frac{5}{2^{2^n}+1}\right)$, when $n>1$ Solution: ...
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Question about semi-twin primes

Has anyone ever studied the following numbers? I came up with the name semi-twin primes as I could not find any reference on this topic. These are prime numbers $p$ such that $p+1 = 2q$, with $q$ ...
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Minimum number of points to win a football league

Let $n$ be the number of teams in a football league. $n$ is even. Each pair of teams plays each other $2$ times (home and away) during the season. There are $2(n-1)$ fixtures and in each fixture there ...
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what is formula for $\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$ and Real Roots Polynomials

Definition let $b$ and $m$ are non negative integers with $b\ge 1$ $$D_{m,b}=\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$$ I observed $$D_{m,m+1}=m!$$ $$D_{m,m}=(m+1)!/2$$ D_{m,m-1}= (m+1)...
Definition: The $n$ successive minima $\lambda_1,..,\lambda_n$ of $C$ with respect to lattice $L$ are defined as follow $\lambda_i$ is the minimum of all positive reals $\lambda$ such that \$\lambda C ...