# Questions tagged [integers]

For questions about the structure, definition, and basic properties of the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$. Do not use this tag just because your question involves integers. Consider using (elementary-number-theory) or (number-theory) instead of or in addition to this tag.

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### Is there a mathematic formula for n+ceil(n/2)+ceil(n/4)+...

Is there a mathematic formula for n + ceil($\frac{n}{2}$) + ceil($\frac{n}{4}$) + $\dots$? I know that $n + \frac{n}{2} + \frac{n}{4} + ... = 2n - 1$. Currently, I'm calculating each term and adding ...
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### Separating Gamma function in two independent functions: $\Gamma(n-m) = f(n)g(m) ?$

I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is Let $\Gamma(z)$ be the gamma function. Given integers $n > m$, ...
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### (sub)monoids of the positive integers under multiplication, with density $0$ in the positive integers, are always multiplicative norms of rings?

Consider integer polynomials of type $"x"$ where we take as imput nonnegative integers. With these nonnegative integer imputs we strictly generate a subset of nonnegative integers ; the set $X$. The ...
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### Integer Solutions to the Equation $(n-1)(x+1)(y+1)(z+1)=nxyz-1$

How can I find all integer solutions for the equation $$(n-1)(x+1)(y+1)(z+1)=nxyz-1$$ for any given positive $n$ where $n≤x≤3n-2$ and $x≤y≤z$? All attempts by me to solve this problem have so far come ...
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### Existence of solutions in diophantine equations

Recently I've wondered if one equation in integers had a solution and faced a stunning (as for me) question. There is a method of proving that equation doesn't have solutions by looking at it over the ...
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### Roots of $x^2-x+2=0 \in \mathbb{Z}_3[i]$

I've been challenged by a professor to find the roots of $x^2-x+2=0$ in the field $\mathbb{Z}_3[i] = \{a+bi \; \vert \; a,b \in \mathbb{Z}_3\}$. I used the "normal" quadratic formula and got ...