Tagged Questions

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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Any Mersenne prime contains two consecutive 9 digits? [on hold]

The kids with me were each asked to pick a number. It crossed my mind that a smart aleck might answer with a description of some number that we have never actually computed. I remembered that a ...
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The number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$

For given positive integers $r,v,n$ let $S(r,v,n)$ denote the number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$ and such that $x_i \leq v$ ...
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Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
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Are the extremas of $h(x)$ global?

It is well known that $li(x)$, the integral logarithm is a very good approximation of $\pi(x)$, the nunmber of primes not exceeding $x$. So, a very good approximation for the probability, that a ...
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$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [on hold]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
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Find the maximum value that the quantity $2m+7n$ can have

Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i$ $(1 \leq i \leq m)$, $y_j$ $(1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$...
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Books for maths olympiad

I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
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Assuming $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1}+a_{n}$ for $n \geq 0$, prove that $\gcd(a_m,a_{m+1}) = \gcd(a_m,a_{m-1})$

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1}+a_{n}$ for $n \geq 0$. Prove that if $\gcd(a_m,a_{m+1}) = d > 1$, ...
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Find the common divisors of $a_{1986}$ and $a_{6891}$

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1, a_{n+2} = 4a_{n+1}+a_{n}$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$. ...
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If $p$ and $q$ are odd, prove that there are no integral solutions

Prove that if $p$ and $q$ are odd numbers , then the equation $x^{10} + p x^{​9} + q = 0$ does not have integral solution. Could some hint a simple approach to solve this question. I am not getting ...
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For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square?

Question. For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square? Clearly, $n=24$ is one such value, and I was wondering whether this is the only value for which the above holds. The ...
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Strange results in mersenne.org database

I am interested in GIMPS project. I was browsing through known Mersenne prime numbers when I discovered strange records in their database. For example, M6972593 is the 38th Mersenne prime. However, ...
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Prove that $\text{ord}_{2^n}(x) = 2^k$

Let $p$ be a prime. Define $v_p(x)$ such that if $v_p(x) = n$, then $p^n \mid x$ but $p^{n+1} \nmid x$. Prove that if $v_2(x^{2^k}-1) = n$ where $\gcd(x,2^n) = 1$, then $\text{ord}_{2^n}(x) = 2^k$. ...
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Distribution of numbers of the form $p^{q-1}$, p and q prime

Prime numbers are exactly the integers having 2 divisors and of course, 2 is itself prime. If one considers the set of positive integers with a prime number of divisors, one can easily figure out that ...
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