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

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12
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502 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 ...
11
votes
0answers
578 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 ...
10
votes
0answers
109 views

Peculiar numbers

This is from the weekly math challenge of the French newspaper Le Monde. "Magical" numbers have a remarkable property: when one of them, say $n$, is squared, it is such that $n^2$ ends with $n$. ...
9
votes
0answers
158 views

When is $(x^n-1)/(x-1)$ a prime number?

Let $x > 1$ and let $n$ be a prime. I'm wondering if a characterization of this is known. That is, what are sufficient and necessary conditions for $$ \dfrac{x^n-1}{x-1} = 1 + x + x^2 + \cdots + ...
9
votes
0answers
161 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 ...
9
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0answers
104 views

infinitely many

It is well known that there are infinitely many positive integers $n$ such that $2^n+1$ is divisible by $n$. Also it is well known that there exist infinitely many positive integers $n$ such that ...
9
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0answers
268 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 ...
9
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178 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 ...
8
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0answers
429 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 ...
8
votes
0answers
579 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
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0answers
151 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
180 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
126 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
66 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) ...
7
votes
0answers
123 views

Show that $x$ is rational.

Sincerely, I don't have the slightest idea for this one : Suppose that $a,b$ are distinct positive integers and that the numbers $\lfloor a^n x\rfloor $ with $n\in\Bbb{N}$ and $x$ a fixed real ...
7
votes
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79 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
198 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 ...
6
votes
0answers
84 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 ...
6
votes
0answers
37 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
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104 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 ...
6
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74 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
156 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
127 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
105 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
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0answers
295 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
208 views

Diophantine special problem

This is my another question on Diophantine equations. Prove the following great and special problem. Let $D$ and $k$ be positive integers and $p$ be a prime number such that $gcd(D, kp) = 1$. Prove ...
6
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320 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
66 views

Find the Number of Integral Solutions

Let $k$ be an integer such that $1 \leq k \leq \left(\dfrac{p-1}{2}\right)$ for some odd prime $p$. Let $ a$ be another integer such that $1 \leq a \leq (p-1)$. Then find the number of integral values ...
5
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0answers
136 views

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > ...
5
votes
0answers
84 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 ...
5
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0answers
79 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
5
votes
0answers
167 views

How to partition $S$ in this way?

Assume: $$ P =\{p_1,p_2,\cdots,p_K\}\subset \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N} $$ and, $$ f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j ...
5
votes
0answers
222 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
5
votes
0answers
114 views

Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For my application, I ...
5
votes
0answers
120 views

divisibility by powers of $2$ of diagonal sums of infinite latin square

This array is formed by placing integers such that each is the smallest such that the rectangle with it and the top left hand corner as opposite corners does not contain the same integer twice in any ...
5
votes
0answers
159 views

Carmichael number factoring

The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further ...
5
votes
0answers
362 views

Digital Numbers using all digits from 1-9

Call a number a digital number if it consists of all the digits from 1-9, each used exactly once. What is the probability that a digital number will be divisible by 7 ? What is the probability that a ...
5
votes
0answers
129 views

Question about a proof on p70 in Cassels' Local Fields

I'm trying to read the proof of COROLLARY. The only solutions of $x^2+7=2^m$ ($x,m \in \mathbb{Z}$) (6.15) have $m=3,4,5,7,15$. I don't see why there could be a + in $y\pm \alpha$ ...
5
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0answers
512 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 ...
4
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0answers
154 views

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
4
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0answers
52 views

Does every integer $n > 2$ have a “reciprocating” pseudoprime $a$?

Such that $\gcd(a, n) = 1$, satisfying both $a^{n - 1} \equiv 1 \mod n$ and $n^{a - 1} \equiv 1 \mod a$? (And of course allowing for the idea of even pseudoprimes). From Fermat's little theorem it ...
4
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0answers
89 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
4
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0answers
231 views

Has this weaker version of Fermat's last theorem already had an elementary proof?

Recently I carried out an elementary proof of the following assertion, which is a special case of Fermat's last theorem: If $p$ is an odd prime and $x, y, z > 0$ are integers such that $(x, y) = ...
4
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0answers
71 views

How to efficiently calculate $ax+b$ once I know $a$ and $b$?

What's the cheapest way to calculate $ax+b$ several times once I know the values for $a$ and $b$? For instance, the cheapest way to calculate $a+b+x$ several times once I know the values for $a$ and ...
4
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0answers
28 views

Where can I find proof - There're infinitely many primes $p$ such that $p(mod\ N)\not\in H$ - Name?

Origin - http://math.uga.edu/~pete/4400FULL.pdf - on p120, Theorem 122 Fix a positive integer $N>2$, and let $H$ be a proper subgroup of $U(N)=(Z/NZ)^{\times}$. There are infinitely many ...
4
votes
0answers
131 views

Consider the equation $b^n=a^b$, where $a,b,n\in \mathbb{N}^{\ast}$

DISCLAIMER: This is a self-made problem, and I don't know if there's a nice solution. Hi, here's a conjecture I have for you; I'm not seeing the "simple" solution (at least for part (a)) at the ...
4
votes
0answers
155 views

start studying advanced topics in number theory

I am a first year undergrad and have had elementary course in Number Theory which includes only basic introductory topics like: divisibility,gcd-lcm,primes,congruences, number theoretic functions etc. ...
4
votes
0answers
96 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 ...
4
votes
0answers
64 views

Intersection between OEIS-A050808 and OEIS-A080053

Is there any result on the intersection between A050808 and A080053: OEIS-A080053 = 1, 2, 4, 5, 6, 10,... (Exp(n) is further from an integer than any previous exp(k)) OEIS-A050808 = 1, 2, 18, ...
4
votes
0answers
113 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...