Questions tagged [integers]

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

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238 views

How to prove that $4x^3+6x^2+4x+1$ is not a fourth power of an integer, for any $x\in\mathbb N$?

How do I prove that for all positive integers $x$ it's true that, $4x^3+6x^2+4x+1$ is not a fourth power of an integer? I've tried doing modulo 3 and 5 checks, and it didn't really go far from there
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2answers
235 views

Induction and Maximum Principle

I wish to show that the following two assertions are equivalent: (Principle of Mathematical Induction) Let $S$ be a nonempty subset of the set of non-negative integers satisfying the following two ...
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1answer
143 views

Prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
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2answers
947 views

Prove that if a and b are positive integers, then there exists integers x and y such that 1/lcm(a,b)=x/a+y/b

My professor has not taught us the technique of writing proofs, he just continues to do them for us in class. So I am really stumped on this proof. Any help is greatly appreciated!
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3answers
54 views

Find real functions knowing their values for all natural point

Consider a function $f(x) : \mathbb{R} \rightarrow \mathbb{R}$ smooth enough such that $f(\mathbb{N}) \subseteq \mathbb{N}$. Is there some methodologies to find another function $g(x): \mathbb{R} \...
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1answer
49 views

A question about numbers with a certain property

Find (if exists) a subset of the non negative integers $X$ such that for every non negative integer $n \in \mathbb{N}\cup\{0\}$ there is exactly one solution of the form $a+2b=n$ with $a,b \in X$ I ...
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3answers
1k views

I am thinking of a two digit number… (System of Equations Question)

I am thinking of a two digit number. If the digits of my number are reversed, the new number is 36 greater than my original number. If the tens digit of my original number is doubled and the units ...
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1answer
65 views

Congruence modulo n where one side is equal to 1

In the Miller-Rabin test for prime numbers, there is a congruence in the form of $a^{n-1} ≡ 1$ (mod $n$). I'm curious as to how $1$ modulo $n$ cannot just be written as $n$? And the left side ...
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2answers
94 views

Calculate number of integers less than n fitting the form 6n±1

Of course the approximation is n/3, but I am looking for a way to get the number of integers, not an approximation.
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2answers
4k views

Solve the equation $a+b+c=abc$ for $a,b,c\in\mathbb{Z}$

Solve for $a,b,c$ (where $a$, $b$, and $c$ are integers) the equation $$a+b+c=abc.$$ I would prefer a solution using trigonometry and I think that it might use the formula $\tan A + \tan B + \tan C=\...
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151 views

$\mathbb{Q}$ can not be embedded in $\mathbb{Z}$

Show that $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$ (where both has the subspace topology of $\mathbb{R}$) My attempt at a solution Since Z is discrete, {k} is open in $\mathbb{Z}$ with ...
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2answers
484 views

Add integers to a set number of lists, so that the sum of each completed list is as closely matching to the other lists as possible?

I am trying to figure out how to solve this problem in computer science. I won't go into the programming side of things, but basically what I need is this: I have a list of integers ranging from ...
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1answer
77 views

Diophantine equation: $2(x^3+xy+y^3)=3(x+y)$

Here is a nice equation: $2(x^3+xy+y^3)=3(x+y)$ over $ \mathbb{Z}$ x $\mathbb{Z}$. Any nice way to approach this?
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0answers
107 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
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2answers
170 views

Encyclopedia of integers

Many years ago I read something that mentioned a book I would like to find. Apparently this book is sort of an encyclopedia for integers; each entry lists interesting mathematical facts about that ...
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3answers
48 views

Equation that can easily be changed to output the digit in 1's, 10's ,100's etc?

I need an equation that can be easily changed to output the digit which is held in the 1's slot, 10's slot, 100's slot, etc. EX. I want the 100's digit in 6810 EX2. I want the 1's digit in 29115 ...
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1answer
62 views

Find a function that maps x,y to $[0, n ( n + 1) / 2)$

Can you find me a bijective function that maps positive integers $x, y$ such that $0 \leq x < y \leq n$ to integers in $[0, n(n+1)/2)$ to use as a hash function?
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1answer
91 views

Integers and place values?

Suppose the tens digit of a whole number between 80 and 90 is greater than the ones digit,but less than twice the ones digit. If the integer is even, what is it's value?
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1answer
4k views

Natural and Real sets of numbers, which one is bigger than another?

From the years ago, it has always been this question in my mind which a teacher of high school talked about in a class but I never found it's correct answer. We have set of natural numbers ${1,2,3,4,...
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2answers
167 views

How solve this: if $\sin{(ax+b)}=\sin{29x}$ for all integers $x$, find the smallest possible value of $a$

Question: Assume that $a$ and $b$ are nonnegative real numbers such that $$\sin{(ax+b)}=\sin{29x}$$ for all integers $x$. Find the smallest possible value of $a$. This problem is from a China book, ...
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2answers
99 views

Solution of an equation involving even integers

If $x$ is any positive even integer $> 1$. I compute: $$ c = x + x! $$ Now assume instead $c$ (even integer) is given, and I want to get back the value $x$. Is it possible to find a simple ...
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2answers
78 views

Distribution of numbers in everyday life

If you were to read tomorrow's newspaper it is intuitively more likely that the whole number 1 would appear more times than 643689443. Is there an expected distribution of numbers used in general? I'...
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3answers
113 views

Is there an explicit polynomial form for the product of consecutive integers?

I have the product $\prod_{j=0}^{r-1} (n+j)= n(n+1)\cdots(n+r-1)$ where n is a positive integer, and I was wondering if there was an explicit polynomial form for it (as a polynomial of degree r). I've ...
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1answer
176 views

Do perfect polynomials of degree $4$ exist?

I asked this question already, but I cannot find it anymore. If it is a duplicate, I will delete it. Is there a polynomial $$p(x)=x^4+ax^3+bx^2+cx+d$$ such that p and all the derivates upto the ...
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2answers
133 views

Sum of floor of ratios

I need to compute, in a program at work, the sum, for $k = 2$ to $n-1$, of the floors of the ratios: $\frac{n}{k}$. Since n is a large integer in my case I would need a "closed form" formula for this ...
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1answer
144 views

A sequence of subsets of $\Bbb Z$ not containing nontrivial subgroups

Is there a sequence $(A_n)$ of subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in A_{n+1}\}$ is a proper subset of $A_n$ and no $A_n$ contains an infinite subgroup of $(\Bbb Z,+)$? (Ed.: this is ...
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1answer
74 views

About a nested sequence of subsets of integers

Let $(H_n)$ be a sequence of nonempty subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in H_{n+1}\}\subsetneqq H_n$. Can we deduce that there is some $n$ such that $\{a-b\mid a,b\in H_{n}\} = H_n$?...
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2answers
725 views

Find a number by the decimal part of its square root [duplicate]

I have a math problem consisting of two questions: can we find a number N knowing only the decimal part of its square root up to a precision (only an approximation of the decimal part because the ...
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2answers
57 views

Proof that if $k$ is the highest factor of any positive integer $n$ such that $k<n$, then $n/k$ is prime

It's straightforward to say that when $n$ is prime, $k=1$ since $k$ must be less than $n$. For the case where $n$ is not prime, I thought proving that the lowest factor of $n$ is prime would be the ...
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1answer
166 views

Integer as a product of a square and a square free integer

My question actually relates to an example given on p. 28 of Julian Havil's "Gamma". Discussing a proof of the infinity of primes due to Erdos, Havil writes: [Erdos] uses a counting technique ... ...
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1answer
237 views

how to tell a fraction in denominator or numerator should be substituted with its integer equivalent

Suppose we have equations as follows (A, C and B are all integers and $\gcd$=greatest common divisor). $$R_1 = \frac{A\times C}{B} \hspace{2cm} R_2 = \frac{A\times\frac{C}{\gcd(B,C)}}{\frac{B}{\gcd(B,...
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2answers
1k views

Why is (European) money in units of $1,2,5,10,20,50, \cdots\;$?

In the old days, in the Netherlands, we had 1 ct (cent), 5 ct (stuiver), 10 ct (dubbeltje), 25 ct (kwartje), 1 gld (gulden), 2.5 gld (rijksdaalder), 10 gld (tientje), ... And then they decided we ...
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1answer
237 views

Count the number of ways n different-sided dice can add up to a given number

I am trying to find a way to count the number of ways n different-sided dice can add up to a given number. For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ($(2,6),(3,5),(4,4)...
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2answers
739 views

Uncountable models for integers

Part of Asaf Karagila's brilliant answer to one of my other questions puzzles me a lot. Namely, I find it hard to understand how there can be a model for ZFC with uncountably many integers. My ...
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1answer
138 views

Finite integral with goniometric functions, $\int_0^{\infty} \frac{8\sin^4(\pi f t)\tan^2(\pi f/2)}{(\pi^4 \tau^2 f^3) }df$

I have difficulties trying to find an algebraic solutions of the following integral: The $\tau$ in this formula is an integer, which is a very important fact because only then this integral is ...
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1answer
161 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
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1answer
2k views

Calculating a Factorial Base Representation

My friend thought of a system in which each number $n$ (I will first restrict my question to positive integers $n$) is represented by a digit string $(d_l,...,d_1)$ as follows $\forall n \in \mathbb{N}...
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1answer
254 views

Proving the Well-Ordering Property

My textbook states the Well-Ordering property as following: If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \le a$, for all $a$ in $A$ ($m$ is called ...
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4answers
70 views

What is the probability that the difference of squares of two positive integers up to $30$ is divisible by $3$ or $7$?

If we choose any two numbers $a$ and $b$ from the integers $1$ through $30$, what is the probability of $a^2-b^2$ of being divisible by $3$ or $7$?
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2answers
92 views

How can I solve: ${\left [{x+1}\over2\right]}={x-1\over 3}$?

How should the following equation be solved? $${\left [{x+1}\over2\right]}={x-1\over 3}$$ where $[a]$ is the integer part of the number.
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1answer
739 views

Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$

I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction but I ...
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3answers
31k views

The best symbol for non-negative integers?

I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable? $\mathbb{N}_0$ $\mathbb{N}\cup\{0\}$ $\mathbb{...
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3answers
163 views

How find this $\lim_{n\to\infty}\left(\frac{1}{a_{n}+1}+\frac{1}{a_{n}+2}+\cdots+\frac{1}{a_{n}+b_{n}}\right)=x$

Prove that for any $x\in[0,\infty)$ there exist sequences of positive integers $\{a_{n}\}_{n\in\mathbb N}$ and $\{b_{n}\}_{n\in\mathbb N}$, such that $$\lim_{n\to\infty}\left(\dfrac{1}{a_{n}+1}+\dfrac{...
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2answers
4k views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
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5answers
2k views

Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each $...
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2answers
9k views

Probability that two random numbers are coprime is $\frac{6}{\pi^2}$

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...

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