# Questions tagged [integers]

For questions about the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$.

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### Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
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### What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact ...
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### Why are integers subset of reals?

In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). ...
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### Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. ...
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### The equation with binomial coefficient $\binom{n-m}{k+m}=\binom{n+m}{k-m}$

Find all positive integers $n,k$ such that $$\binom{n-m}{k+m}=\binom{n+m}{k-m}$$ 1) I solved problem if $m=1$. Its here: $k=1; n=3$ 2) $$\binom{n-m}{k+m}=\binom{n+m}{k-m}$$ $k=m, n=3m$ is root of ...
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### How can I find integers which satisfy $\frac{150+n}{15+n}=m$?

Here are some facts about myself: In 2017, I was $15$ years old. Canada, my country, was $150$ years old. When will be the next time that my country's age will be a multiple of mine? I've toned ...
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### proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove ...
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### Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
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### Does the list of “number of groups of order $n$” contain every natural number?

In other words: For every natural number $m$, does there always exist an $n$ for which there are exactly $m$ groups of order $n$ up to isomorphism? Or is this an open question in mathematics? If ...
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### What is the difference between natural numbers and positive integers?

I was reading sets and came to some reserved letters for a few sets. Two of them really confused me. They were - $\mathbb N$ : For the set of natural numbers. $\mathbb Z^+$ : For the set if all ...
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### Proof of Lemma: Every integer can be written as a product of primes

I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is. I encountered the classic lemma about every nonzero integer being the product of primes in Ireland and Rosen's ...
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### Can $\sqrt{ab}$ and $\sqrt{a^2 + b^2}$ be both integers if $a$ and $b$ are natural numbers?

Does there exist an $a \in \mathbb{N}$ and $b \in \mathbb{N}$ such that $\sqrt{ab} \in \mathbb{Z}$ and $\sqrt{a^2 + b^2} \in \mathbb{Z}$?
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### How to find all polynomials that map integers to integers?

How can one find all polynomials of degree k that map integers to integers? In other words, how to get all combinations of coefficients $a_0,...,a_k \in \Bbb R$ ...