Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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5
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3answers
93 views

The only positive integers that divide successive numbers of the form $n^2+3$ are $1$ and $13$

I stuck with this problem, I don't know how to start with. Prove that the only positive integers that can divide successive numbers of the form $n^2+3$ are 1 or 13.
1
vote
3answers
416 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
0
votes
0answers
37 views

On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

(Note: I have edited this question to conform to the further details added in the cross-post to MO.) Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if ...
3
votes
1answer
89 views

Let $x$ be a real number. Prove the existence of a unique integer $a$ such that $a \leq x < a+1$

Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$ ...
-2
votes
4answers
19 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

(Note: This question has been cross-posted from MO.) Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) ...
9
votes
6answers
214 views

Which of the numbers $300!$ and $100^{300}$ is greater

Determine which of the two numbers $300!$ and $100^{300}$ is greater. My attempt:Since numbers starting from $100$ to $300$ are all greater than $100$. But am not able to justify for numbers between ...
-2
votes
1answer
102 views

Equation involving floor function: [closed]

Given n a natural number, find $x$ (positive real number) such that: $$ 6\lfloor x \rfloor=n, $$ where $ \lfloor x \rfloor $ represents the value of the floor function in x.
4
votes
2answers
96 views

How to find out if a number is a hundred or thousand?

The question might raise people's eyebrows but I have been googling and I don't know the keyword to search for. I just don't know the mathematical term. What I'm trying to do is I want to round a ...
3
votes
5answers
90 views

What will be the remainder when $2^{31}$ is divided by $5$?

The question is given in the title- Find the remainder when $2^{31}$ is divided by $5$. My friend explained me this way- $2^2$ gives $-1$ remainder. So,any power of $2^2$ will give $-1$ ...
0
votes
3answers
61 views

Minimizing the difference between two prime numbers [closed]

I tried to somehow simplify the statement to find $(q-p)$ but I couldn't. Here is the question: Given that $p$ and $q$ are prime numbers with $p<q$ and given that $pq-p-q = 59$, what is the ...
1
vote
0answers
79 views

Why is proving that $10$ is solitary considered very difficult?

The title says it all. We denote the sum of the divisors of $x$ by $\sigma(x)$. The ratio $I(x)=\sigma(x)/x$ is called the abundancy index of $x$. If $I(m)=I(n)$, then $\{m,n\}$ is called a ...
0
votes
0answers
25 views

Can distinct odd perfect numbers $N = {p^k}{m^2}$ share the same Euler factor $p^k$?

(A similar question has been asked in MO.) Let $\sigma(x)$ denote the sum of the divisors of $x$, and call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A number $N$ is called ...
13
votes
3answers
756 views

Why is there no general form for the harmonic numbers?

The Harmonic numbers $H_n$ are given by the sum of the reciprocals of the natural numbers up to a given $n$, ie: $H_1 = 1$ $H_2 = 1 + 1/2 = 3/2$ $H_3 = 1 + 1/2 + 1/3 = 11/6$ $H_n$ for noninteger ...
4
votes
2answers
64 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
25
votes
6answers
538 views

What would Gauss do in this case: adding $1+\frac12+\frac13+\frac14+ \dots +\frac1{100}$?

We all know the story related to Gauss that Gauss' class was asked to find the sum of the numbers from $1$ to $100$ as a "busy work" problem and and he came up with $5050$ in less than a minute. He ...
3
votes
1answer
47 views

A relation related with odd perfect numbers

It is easy to prove, using the relation $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ holds for $n\geq 1$ where $\sigma_0(n)$ is the number of divisors, the following Proposition. The integer $n\geq 1$ is ...
131
votes
19answers
10k views

Mental Calculations

This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky. The problem presented on a blackboard requires computing the ...
23
votes
5answers
1k views

Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
5
votes
2answers
104 views

Number of divisors of the form $(4n+1)$

Find the number of divisors of $$2^2\cdot3^3\cdot5^3\cdot7^5$$ which are of the form $(4n+1)$ I know how to find the total number of divisors. But, to find the number of divisors of the form ...
2
votes
4answers
63 views

the product of an odd perfect number and some even perfect number is perfect

If $a$ were an odd perfect number ,does there exist an even perfect number $b$ such that $ab$ is a perfect number?
1
vote
1answer
86 views

Show that $xyxyxy$ is not a perfect power.

If $N=xyxyxy$ where $x$ and $y$ are digits. Show that $N$ cannot be a perfect power, i.e. $N\ne a^b$, where $a$ and $b$ are positive integers and $b>1$. My work $xy|xyxyxy$ and ...
6
votes
2answers
158 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
1
vote
1answer
38 views

Legendre Symbol $\left(\frac4p\right)$ is always congruent to $1$?

Let$\newcommand\leg[2]{\left(\frac{#1}{#2}\right)}$ $\leg ap$ denote the Legendre symbol. In all cases $a=4$. and $p$ takes values of different odd prime numbers $p$. For $p=5$: $\leg 45$ -> ...
2
votes
2answers
95 views

Proving $2^{2^n}+3^{2^n}+5^{2^n}$ is divisible by $19$ for all $n\geq 1$ by induction

I came across the following in the book Handbook of Mathematical Induction: $$ 19\mid (2^{2^n}+3^{2^n}+5^{2^n}),\quad n\in\mathbb{Z^+}\tag{1} $$ Apparently, this problem is not so bad if you think ...
2
votes
0answers
80 views

Reasoning about the number of elements in a reduced residue system relative a primorial

Let $R_{p_i\#}$ be the reduced residue system relative the primorial for the $i$th prime. Let $\left|R_{p_i\#}\right|$ be the number of elements in this set. It is well known that: ...
3
votes
1answer
81 views

Does every digit occur with equal frequency in the set of prime numbers?

Does every digit occur with equal frequency in the set of prime numbers? More precisely, let $f(n)$ be the total number of base-$b$ digits contained in the first $n$ prime numbers and let $f_d(n)$ be ...
-4
votes
1answer
74 views

Is there is a proof that the events of occurring of twin primes are independent?

The motivation to this question can be found in: http://curvebank.calstatela.edu/prime/prime.htm I am interested on the meaning of the second paragraph: Until recently, it had been conjectured that ...
1
vote
3answers
74 views

Prove $(n!-1,(n-1)!-1)=1$

Question: Let $n\geq2,n\in\mathbb{N}$. Prove $(n!-1,(n-1)!-1)=1$ I have noticed that $n!=n\cdot (n-1)!$ So letting $\alpha=(n-1)!$, we have to prove $(n\alpha-1,\alpha-1)=1$ I feel that this is ...
11
votes
1answer
84 views

Does it follow that $(n!)^n$ divide $(n^2)!$

It is well known that $(n!)^2$ divides $(2n)!$. Does it follow that $(n!)^3$ divides $(3n)!$ and so on up to $(n!)^n$ dividing $(n^2)!$? If yes or no, could you provide the details behind the ...
9
votes
1answer
841 views

Three variable, second-degree symmetric Diophantine equation

Find integers $f,g,h$ such that $3(f^2+g^2+h^2)=14(fg+gh+hf)$. You can do it using a computer or by hand. I tried this problem for ages, got nowhere. Unfortunately I don't know how to program, but I ...
0
votes
1answer
59 views

Hard polynomials problem

Let $n$ be a given positive integer. Find an integer $m$, and polynomials $f,g,h$ in $n$, such that $f^2+g^2+h^2=135m^2n^2-30m^2n-5m^2$ or prove that it is impossible. I've tried this problem for ...
0
votes
1answer
89 views

Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
1
vote
0answers
75 views

Reasoning about prime counting and the twin prime conjecture

I've been thinking the primorial for say the $i$th prime $p_i$and the equations for counting the number of elements in the reduced residue system for this primorial and counting the number of elements ...
3
votes
4answers
73 views

Prove that $(n^2-1)\mid(n^3+1)$ iff $n=2$

Seperating $n^2-1$ into $(n+1)(n-1)$. I have noticed that $n^3+1=(n+1)(n^2-n+1)$, so we have $\forall n\geq 2$, $(n+1)\mid(n^3+1)$. We now need to show that $(n-1)\mid(n^2-n+1)$ iff $n=2$ This ...
0
votes
0answers
37 views

Modular Division and Factorial

I am unfamiliar with number theory but am trying to calculate the following for a coding challenge: $$\frac{(N-M-1)!}{N!(M-1)!}\pmod{Q}$$ where $Q$ is prime. I know that I can calculate the ...
64
votes
12answers
8k views

Is there something special about 2015? [closed]

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...
0
votes
3answers
9k views

Is zero an even number? [duplicate]

A quick google returns the answer on the parity of zero: Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. The simplest way to prove that zero is ...
3
votes
0answers
96 views

Number Theory: Legendre Symbols

I have the following question. Calculate the Legendre symbol $\bigl(\frac{77}{5^{200}+1}\bigr)$. I know the following: $5^6\equiv1\pmod7$, $5^{10}\equiv1\pmod{11}$. Thus I approached this ...
71
votes
11answers
13k views

Am I just not smart enough? [closed]

When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also ...
52
votes
4answers
3k views

Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
4
votes
0answers
195 views

The n-envelope problem

This is original problem: You have n number of envelopes, and $100$ $\$1$ bills. you have to put these bills in the envelopes in such a way that any amount between $1$ to $100$ can be reached just ...
2
votes
1answer
143 views

Estimate or calculate the number of digits of a multiplication

I would like to calculate the number of digits of these multiplications 2 x 4 200 x 300 2 (12321) (1000). I don't exactly know how to start. Of course I know that I can multiply the numbers and ...
2
votes
0answers
44 views

Why is the Legendre symbol $(\frac a p)$ not defined if $p | a$ in some books?

Why is the Legendre symbol $(\frac a p)$ not defined if $p | a$ in some books ? In some textbooks I've come to notice that the legendre symbol $(\frac a p)$ is not defined if $p | a$, where $p$ ...
21
votes
3answers
2k views

Fibonacci numbers that are powers of 2

Are there infinitely many Fibonacci numbers that are also powers of 2? If not, which is the largest?
7
votes
1answer
391 views

What is the period of this sequence?

Consider the recurrence relation: $$x_{i+1}=p-1-((p \cdot i-1) \mod{x_i})$$ If $p$ is prime and $x_0=1$, what is the least period of the resulting (eventually periodic) sequence? My guess is the ...
8
votes
0answers
268 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
2
votes
4answers
1k views

Last digits number theory. $7^{9999}$?

i have looked at/practiced several methods for solving ex: $7^{9999}$. i have looked at techniques using a)modulas/congruence b) binomial theorem c) totient/congruence d) cyclicity. my actual desire ...
12
votes
2answers
1k 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.
32
votes
3answers
1k views

Why is $(2+\sqrt{3})^{50}$ so close to an integer?

I just worked out $(2+\sqrt{3})^{50}$ on my computer and got the answer $39571031999226139563162735373.999999999999999999999999999974728\cdots$ Why is this so close to an integer?
2
votes
2answers
103 views

Prove that $f(n)$ is the nearest integer to $\frac12(1+\sqrt2)^{n+1}$?

Let $f(n)$ denote the number of sequences $a_1, a_2, \ldots, a_n$ that can be constructed where each $a_i$ is $+1$, $-1$, or $0$. Note that no two consecutive terms can be $+1$, and no two ...