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

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27
votes
0answers
722 views

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ ...
22
votes
0answers
1k views

A very nice divisibility problem

A very hard problem, here it is: Prove that, if $2^{2^j} a + 1$ divides $c^{2^j}+1$ for fixed integers $a,c$ and all nonnegative integers $j$, then $a=1$ and $c=2^l$ for some odd positive integer ...
13
votes
0answers
325 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
12
votes
0answers
625 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
11
votes
0answers
377 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
10
votes
0answers
332 views

A question about the divisibility of sum of 2 consecutive primes.

Well as I was curious about the sum of $2$ consecutive primes, after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least ...
10
votes
0answers
221 views

An infinitude of “congruence condition” primes?

Background: Several special classes of primes can be written as primes that satisfy some additional constraint $f(p)\equiv 0\pmod p$; for instance, Wilson primes are congruence primes with ...
9
votes
0answers
131 views

Find all integer solutions of the equation

Find all integer solutions to $3^a+7=2\cdot5^b$. Basically I've tried almost every tool I know of NT (Zsigmondy,LTE,reducing to various modulus) but nothing worked. So far I just know that (1,1) ...
9
votes
0answers
520 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
9
votes
0answers
756 views

How many integer solutions to a diophantine equation

Starting with the equation: $\frac{1}{a}+\frac{1}{b}=\frac{p}{10^n}$, I reached the equation: $10^{n-log(p)} = \frac{ab}{a+b}$. Now given the positive integer $n$, for what integer values of $p$ ...
8
votes
0answers
109 views

Given the first $n$ primes, find the least common multiple of $p_1 - 1$, $p_2 - 1$, …, $p_n - 1$

Given the first $n$ primes, we can label the $k$th prime as $p_k$. So, what is the least common multiple(LCM) of {$p_1 - 1$, $p_2 - 1$, $p_3 - 1$, ..., $p_n-1$}? In other words, if we subtract $1$ ...
8
votes
0answers
231 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 ...
8
votes
0answers
124 views

Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This ...
8
votes
0answers
231 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $x^3-y^5=(x+y)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
8
votes
0answers
184 views

Is there some exact form of $n$ for the number of $(i,j,k)$ satisfying $ijk = (n-i)(n-j)(n-k)$

For any positive integer $n$, $i,j,k$ are also positive integers, and $0 <i,j,k < n$. How many solutions of the form $(i,j,k)$ are there for the equation $ijk = (n-i)(n-j)(n-k)$?
8
votes
0answers
200 views

How many numbers of the form $p_1^2 p_2 p_3$ are there less than $10^{15}$ for $p_1$, $p_2$, $p_3$ distinct primes?

Is there an easy way to compute the following question: How many numbers of the form $p_1^2 p_2 p_3$ are there less than $10^{15}$ for $p_1$, $p_2$, $p_3$ distinct primes? The only thing that ...
7
votes
0answers
127 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
7
votes
0answers
156 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
7
votes
0answers
63 views

Arrange Relatively Prime Numbers in a Circle

The question: In how many ways can you arrange the numbers $1$ to $8$ in a circle so that neighboring numbers are relatively prime? Can you generalize for $1$ to $n$? It's fairly easy to list ...
7
votes
0answers
120 views

Let $p$=prime and $\sqrt{x}+\sqrt{y}<\sqrt{2p}$

Let $p$ be a fixed odd prime. Let $x,y\in \mathbb{Z}_+$ such that $\sqrt{x}+\sqrt{y}<\sqrt{2p}$. Prove that $$\sqrt{x}+\sqrt{y}\le \sqrt{\frac{p-1}{2}}+\sqrt{\frac{p+1}{2}}.$$ Any ideas at all? ...
7
votes
0answers
101 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
7
votes
0answers
182 views

when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
7
votes
0answers
160 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
7
votes
0answers
192 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
7
votes
0answers
158 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
7
votes
0answers
104 views

Which digit occurs most often?

Is there any method to calculate, which digit occurs most often in the number $$4 \uparrow \uparrow \uparrow \uparrow 4\ ,$$ the fourth Ackermann-number ? Or would it be necessary to calculate the ...
7
votes
0answers
309 views

Prove that Bezout's coefficients given by the extended Euclidean algorithm are minimal in absolute value

Bezout's lemma states that if a and b are integers, and at least one of them is non-zero, then there exist integers $x, y$ such that $$ax + by = gcd(a, b)$$ One way of finding such a pair $(x, y)$ is ...
7
votes
0answers
343 views

A conjecture about mersenne primes and non-primes

Latest Edit: My hypothesis is simply that A056295(n) are spread around some smooth curve and higher than the curve if M(n) is a non-prime (and perhaps also even higher yet if M(n) has many factors ...
6
votes
0answers
80 views

A bound on the nth prime.

Is there any combinatorial argument to show that the nth prime $p_n = \mathcal{O}(n^k)$ for fixed $k$ ? There is a problem in the book by Apostol to find upper bounds on $p_n$, the Prime Number ...
6
votes
0answers
174 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
6
votes
0answers
94 views

An Impossible Sequence of Prime Powers

Let $x_1,x_2,\ldots$ be a sequence of positive integers that satisfies the recurrence relation $$x_{n+1}=2x_n(x_n-1)+1$$ for all positive integers $n$. It seems impossible that every term in this ...
6
votes
0answers
132 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
6
votes
0answers
491 views

2015-related question: why are Lucas-Carmichael numbers named after Lucas?

Summary 2015 is a so called Lucas-Carmichael number. I believe (for reasons that I will explain below) that the 'Carmichael' in the name is a reference to ordinary Carmichael numbers and not to the ...
6
votes
0answers
115 views

What is known about the solutions to $\varphi(a)+\varphi(b)=\varphi(a+b)$?

As of late I have been researching Euler's Totient function. For the last week or so I have specifically been studying the equation: $\varphi(a)+\varphi(b)=\varphi(a+b)$ While the equation ...
6
votes
0answers
44 views

Inverse image of rationals under tangent function is free abelian?

It is easy to see that the set $\{x:\tan x\in \Bbb Q \,\, or\,\, \pm\infty\}$ forms a group under addition. It is a free abelian group?
6
votes
0answers
83 views

Does there exist a positive $k$ s.t. for all $r\geq k$, “$\sigma_r(m)<\sigma_r(n)$ for every $m<n$” for infinitely many odd positive integers $n$?

Does there exist a positive real $k$ such that for all real $r\geq k$, "$\sigma_r(m)<\sigma_r(n)$ for every $m<n$" for infinitely many odd positive integers $n$? $\sigma_r(n)$ is the sum of the ...
6
votes
0answers
301 views

number of quadratic residues modulo n

Define $f(n)$ to be the number of quadratic residues modulo $n$. I would like to show that $f$ is multiplicative, that is, for any positive integers $m,n > 1$, $f(mn) = f(m)f(n)$ whenever $(m,n) = ...
6
votes
0answers
148 views

Twin Prime Powers

What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)? I'm aware of the cases for when they are ...
6
votes
0answers
121 views

What is the Gauss sum equivalent of $\Gamma(s+1) = s\Gamma(s)$?

Gauss sums are analogous to the Gamma function: fix a complex number $s$ with real part $>0$. Then we have a multiplicative character $\chi_s :\mathbf R^{\times}_{>0} \to \mathbf C^\times$ given ...
6
votes
0answers
581 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
6
votes
0answers
356 views

On the equation $m^3-m^2+1 = n^2$

(i) How can I find all positive integers $m$ such that $m\equiv 4 \pmod 7$ and $m^3-m^2+1$ is a perfect square? (ii) Is there a method to solve this equation over positive integers: $$m^3-m^2+1 = ...
5
votes
0answers
43 views

Sequence involving floor function - limit and bounds

My first time here, so please excuse any breaches of etiquette. For a given $p \in \mathbb N$ and irrational $\alpha$, let $\varepsilon_n=\alpha n-\lfloor \alpha n \rfloor,$ and ...
5
votes
0answers
53 views

a new(?) operation using products of multiplicities

Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere? Notice that $\odot$ is ...
5
votes
0answers
75 views

Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
5
votes
0answers
76 views

Weakening Goldbach hypothesis by allowing finitely many composites and 1

Is it an open question whether there is a finite set $N$ of positive integers such that for every positive even integer $n$ there are $n_1,n_2\in\mathbb P\cup N$ such that $n=n_1+n_2$? ($\mathbb P$ ...
5
votes
0answers
202 views

Proving $11! + 1$ is prime

Prove that: $$11! + 1$$ is a prime number. Without computing the number (or factorial). Obviously, from Wilson's theorem, a number $n$ is prime if, $$(n-1)! + 1 \equiv 0 \pmod{n}$$ Since $n = ...
5
votes
0answers
108 views

Is the Wikipedia article on the proof for Bertrand's Postulate correct?

I was checking out the Wikipedia article on the proof of Bertrand's Postulate. For Lemma 4, the argument is made for $x\ge 3, x\#<2^{2x-3}$ Here's the proof by induction: $n = 3$: $n\# = ...
5
votes
0answers
143 views

Primes as sum of squares.

If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$, show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares. It is well ...
5
votes
0answers
86 views

number at the circumference

Determine all natural numbers $n$ such that the numbers $1,2,3, . . . ,n$ can be placed on the circumference of a circle, such that for any natural number $s$ with $1 \le s \le \frac{n(n+1)}{2}$ there ...
5
votes
0answers
141 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...