# 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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### prove that $x^2-2y^2=1$ has infinite integer solutions. [duplicate]

Prove that $$x^2-2y^2=1$$ has infinite integer solutions. I know this is a case of n=2 in Pell's equation. But the thing is we are not allowed to use that as a reference in this problem. And even when ...
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### Prove that $x+y = (x-y)^2$ has infinite integer solutions

Prove that $$x+y = (x-y)^2$$ has infinite integer solutions. I tried to reform the equation in several ways. As $$(x-y)(x-y-1)=2y$$ Or $$(x+y)(x+y-1)=4xy$$ I was trying to find y in terms of x But as ...
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### Formalizing daily use of integer representation

In daily life, it is common practice to use a sequence of number elements as an integer, e.g., 999 is a decimal number. As I am reading rigorous construction of numbers from Rudin's mathematical ...
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### Troubles with proving R is an equivalence relation [duplicate]

With $A = \mathbb{Z}$ and $B = \mathbb{Z}-\{0\}$, I'm trying to prove that the relation $R$ defined on $A\times B\,$ by $(a,b)R(c,d)\,$ iff $\,ad=bc$, is an equivalent relations. While I do see why ...
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### Does for every $n>1$ exist a sum of $n$ squares of consecutive natural numbers, that equals to the sum of squares of next $n-1$ consecutive numbers?

Can we, for every natural $n>1$, find such a natural number $k$ that sum of $n$ squares of consecutive natural numbers starting with $k$, that is $k^2+(k+1)^2+...+(k-1+n)^2$, will be equal to the ...
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### If $n>m>0, n, m \in \Bbb{N}$, is $n^m-m^n$ positive or negative? (Explain with cases if needed.)

If $n>m>0, n, m \in \Bbb{N}$, is $n^m-m^n$ positive or negative? (Explain with cases if needed.) First, for finding patterns, I've put some small integers to $(n, m).$ \begin{align} (2, 1): \; &...
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### Proving no integer solution exists that makes a polynomial a perfect square

The context for this is the following coding problem on Hackerrank. I'm trying to understand why one of their sample inputs (Sample Input 4) has no solution. After a bit of math, it comes down to ...
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### Why is it possible to define a partition of integers as prime, composite and {1,-1,0} although divisibility is not an equivalence relationship?

It's a known theorem that if $\mathcal{R}$ is an equivalence relation defined on a set, let's say $A$, then $\mathcal R$-equivalence-classes define a partition of $A$. It is also known that the ...
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### Possible to determine the sign based on individual values?

I have a list of positive and negative integers and I wondered if there was a way to determine if the total of those values would be positive or negative, without actually summing the individual ...
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### Queen's graph diameter and knight's graph diameter for a $n \times n \times \dots \times n$ chessboard

Let a $n \times n \times n$ chessboard be given. I have just proven (by brute force) that, the queen's graph diameter, ${d_{n}^k}(Q)$ (i.e., the number of moves needed to move a queen from any 3D cell ...
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### For what integers $n$, do we have $30n+11=6x^2+5y^2$ for some integers $x,y$?

I am trying to find all integers $m$ such that $m$ is relatively prime to 30, and $m=6x^2+5y^2$ for some integers $x,y$. Note that we must have: $y$ is odd, $(y,3)=1=(x,5)$. Using these conditions, I ...
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### Is there a formal approach to rounding of numbers?

Math formalizes arithmetic as algebraic structures and operations on those structures. Rounding numbers is a common operation. Is there a commonly-accepted formalization of that operation as well? E.g....
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### Exact fast factorial computation

For calculating a power, i.e. $a^b$, well-known fast algorithm exists, based on bitwise decomposition of $b$ and sequential squaring of $a$. Are there similar fast algorithms for calculating a ...
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### One type of integer divisor homology - what does this one measure?

The set of functions $\{ (d \mid \cdot) : d \in \Bbb{N}\}$ where $$(d\mid x) = \begin{cases} 1, \text{ if } d \mid x, \\ 0, \text{ if } d \nmid x \end{cases}, \ \forall x \in \Bbb{Z}$$ are the "...
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### Find all positive integers x, y

Find all positive integers x, y such that $(x^2 + y)(x + y^2) = (xy)^3$ I get the problem from a practice problem set of a math camp which was organised for our national math olympiad. I think that ...
Find all positive integers $x, y, z$ such that $x^2y+y^2z+z^2x = 3xyz$. I first tried to solve it by spilting $xyz$ to every expression and factor it. But it fails. I notice that it is true for $\... 0 votes 0 answers 34 views ### A problem of integer approximation Suppose we are given k integers n1, n2, ..., nk and a "target" integer m. We are required to get as close as possible to m using only the given k integers (each one only once) combined via ... • 279 4 votes 0 answers 69 views ### Square numbers with a form of$p^{2n+1} + q^{2n+1}$, where p, q are prime numbers and n is a positive integer Recently I came across a difficult problem: Let$n$be a fixed positive integer and$p, q$be different prime numbers. Can$p^{2n+1} + q^{2n+1}$be a square number? When I wrote a python program, I ... 1 vote 2 answers 113 views ### For a given sequence consisting of a fraction of 2 functions, is there a way to know how many entries are of integer value? [duplicate] For a given sequence$\frac{n^2+k^2}{2n+1}$where$k$is a given integer and$k > 0$, is there a way to calculate how many entries will be of integer value. Or, if that is not possible, know if it ... • 37 1 vote 1 answer 54 views ### If$B$is an integer matrix, does$\mathrm{det}(B^TB)$has to be a perfect square? Suppose$B \in \mathbb{Z}^{m \times n}$be an integer matrix of rank$n$. I want to prove that$\mathrm{det}(B^TB)$is a perfect square. Of course if$m=n$,$\mathrm{det}(B^TB) = \mathrm{det}(B)^2$... • 12.3k 0 votes 1 answer 54 views ### What is an upper bound of$(x - r)_{\pmod d} + (x + r)_{\pmod d}$in terms of$x$and$d$? Let$x, r, d \in \Bbb{Z}$,$x,d \geq 1$and$r \geq 0$. I'm wondering what an upper bound for the expression: $$f(x,r,d) = (x - r)_{\pmod d} + (x + r)_{\pmod d}$$ is in terms of$x, d$only. I ... • 19.8k 1 vote 1 answer 256 views ### Find$a_n$such that the formula is equal to$\zeta(2)$My question start with the observation : $$\sqrt{e}\simeq \frac{\pi^2}{6}$$ At first glance it's not really convincing but after some work I found : $$\sqrt{e-5\left(\frac{1}{\pi}-\frac{1}{e}\right)^{... • 3,317 1 vote 0 answers 8 views ### Computing polynomial expression with a bounded space for intermediate results Consider the problem of computing a general numeric expression over integer variables, made of sums, products, products by a constant and powers. E.g. something like (2xy^3 + 4y)(2z + 5)^3. However, ... • 163 1 vote 1 answer 50 views ### Is there a specific name for this counting system? I needed a counting system that goes like in the spreadsheet below. The columns are different representations of the same counting system. Given a limit of X digits (5 in this example), we count by ... • 1,054 0 votes 1 answer 40 views ### There are 45 multiples of 3 between 85 and integer b. Why is the largest possible value of b not 219? There are 45multiples of 3 between 85 and integer b. What is the largest possible value of b. (x-87)/3 + 1 = 45 \implies x=219 Why is the answer 221? 1 vote 1 answer 47 views ### Definition of integers in higher-order logic There's the classical of natural numbers in higher-order logic (see the introduction of this page for example). Is there something similar for integers (elements of \mathbb{Z}) ? I didn't find this ... • 11 0 votes 1 answer 32 views ### A formal analytical way to solve system of algebraic equations within integer domain Let's take this famous puzzle as an example. I can write it as a system of equations like this$$ 3(100C+10A+R)=(100R+10R+R)  C \in \mathbb{Z}  A \in \mathbb{Z}  R \in \mathbb{Z}  ... • 151 0 votes 0 answers 34 views ### May I know why the two formula is equivalent? This is a lemma 6 in published article called The Number of Prime Factors on Average in Certain Integer Sequences, page8. i have some trouble expounding or looking for the in between equation, before ... • 103 1 vote 1 answer 35 views ### Is it possible to simplify$n \equiv - \frac d 4\bigg(\frac 3 2\bigg)^{m-2} - 2 \pmod {3^m}$further? Is it possible to simplify this further: $$n \equiv - \frac d 4\bigg(\frac 3 2\bigg)^{m-2} - 2 \pmod {3^m} \qquad \text{where } n, d \geq 0 \text{ and } m \geq 2 \text{ are integers}$$ or to glean ... • 1,381 1 vote 1 answer 37 views ### For which integer values of$m \geq 0$and$n \geq 0$is$A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$a positive integer? For which integer values of$m \geq 0$and$n \geq 0$is$A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$a positive integer? I made a table of the first few expressions for$m \in [0, 5]\begin{align} m &... • 1,381 6 votes 3 answers 244 views ### Prove that 2\cdot 3^x +1= p^y has no solution Prove that the Diophantine equation of 3 variables (x,y,p)2\cdot 3^x +1= p^y$has no solution where$x,y\in\mathbb{N}_+$,$x\ge2, y\ge2$and$p$is a prime number. I found that$y\$ cannot be ... 