Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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1answer
370 views

Does Hensel's lemma give ALL solutions to a congrence equation?

Solve $x^{15} \equiv 6 \pmod{7^2}$. My approach based on Hensel's lemma: First let's solve $x^{15} \equiv 6 \pmod{7}$, observe $3$ is a primitive root $\pmod{7}$, so let $x=3^y$ to get $15y \equiv 3 ...
0
votes
2answers
382 views

Proving Gauss' polynomial theorem

Let $P \in \mathbb{Z}[x], P(x) = \displaystyle\sum\limits_{j=0}^n a_j x^j, a_n \neq 0$ and $a_0 \neq 0$; if $p/q$ is a root of P (with p and q coprimes) then $p|a_0$ and $q|a_n$ I've managed to prove ...
-1
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3answers
165 views

Is this possible to convert one array to another given array?

you are given two arrays having n elements , like for n=4,suppose array1={1,2,3,4} array2={2,1,4,5} convert array 1 to array2 performing operation minimum number of time . Also state if ...
-1
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1answer
214 views

Non-trivial automorph of an indefinite form

EDIT of the question by Will Jagy: Is the binary quadratic form $x^2 - 3 y^2$ reduced? What are its proper integral automorphs?
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2answers
270 views

Primes of the form $a^2+qb^2$

Now I came with some very interesting results. Take $p = a^2 + qb^2$ with p is some odd prime and a, b are some integers. Then, (1) Fixing q = 10, p = m (mod 40) for m belongs to the set of 1, 9, ...
-13
votes
1answer
898 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
14
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1answer
157 views

Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...
12
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2answers
238 views

An upper bound for $\log \operatorname{rad}(n!)$

Let $n>1$ be an integer and let $\operatorname{rad}(n!)$ denote the radical of $n$-factorial. (The radical of an integer $m$ being, loosely speaking, the product of the prime divisors of $m$.) Can ...
11
votes
1answer
457 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
10
votes
2answers
285 views

Given the first N integers, how many large prime factors can I disallow and still have half the set remaining?

Conjecture: For $N$ sufficiently large, take the set of positive integers up to $N$. Then remove all numbers which have a prime factor larger than $\sqrt{N}$. More than half the set will remain. ...
9
votes
2answers
247 views

Source of Hardy-Littlewood's 2nd Conjecture

In what paper do Hardy and Littlewood first mention, specifically, their 2nd conjecture? It is not mentioned specifically in Partitio Numerorum III. This conjecture is usually expressed as ...
9
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1answer
174 views

What does this music video teach us about 863?

This delightful animation by Stefan Nadelman depicts "the additive evolution of prime numbers", set to Lost Lander's song "Wonderful World": http://www.youtube.com/watch?v=TZkQ65WAa2Q. (If you haven't ...
9
votes
5answers
343 views

How can I compute the sum of $ {m\over\gcd(m,n)}$?

$$ \sum_{m =1}^n {m\over\gcd(m,n)}$$ For example, for 1 it is $${1\over\gcd(1,1)} =1;$$ for 5 it is $${1\over \gcd(1,5)}+{2\over \gcd(2,5)}+{3\over \gcd(3,5)}+{4\over \gcd(4,5)}+{5\over ...
9
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2answers
298 views

Efficient method to compute $\mathrm{gcd}(2^n-1,n!)$

How can we compute the value of $\mathrm{gcd}(2^n-1,n!)$ efficiently where $n$ is very large? I couldn't think of any fast and efficient method.
8
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2answers
352 views

How often is a sum of $k$ consecutive primes also prime?

Let's define a $k$-sum as a sum of $k$ consecutive primes. For example, $15=3+5+7$ is a $3$-sum. How many $k$-sums are themselves prime? Here's one way to formulate the question more precisely: What ...
8
votes
3answers
142 views

Remainders of binomial coefficients.

Motivation: It is easy to notice that a polynomial map $f: \mathbb{Z} \to \mathbb{Z}$ does not need to have only integer coefficient. For example, $f(x) = \frac{x(x-1)}{2}$ does have rational ...
8
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3answers
3k views

Find maximum divisors of a number in range

While playing around with programming, a problem suddenly came across my mind. Given $n$ let's say $2^{64}$ (64-bit unsigned integer). Find a positive integer $x$, $1 \leq x \leq n$, such that ...
8
votes
1answer
849 views

Do we have a proof of the infiniteness?

Crossposted on Mathoverflow. Given a natural number $a$, are there infinitely many natural numbers not of the form $anm \pm m \pm n$, $n, m$ positive natural? I give a proof that for $a=6$ the ...
8
votes
1answer
560 views

0.246810121416…: Is it a algebraic number?

Is it algebraic the number 0.2468101214 ...? (After point, the natural numbers are juxtaposed pairs).
8
votes
2answers
733 views

Binomial coefficients: how to prove an inequality on the $p$-adic valuation?

In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used: $$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...
7
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4answers
879 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
7
votes
3answers
426 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
7
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1answer
119 views

Need help on proving limit

For each $k$, $$\sum _{n=k^2+1}^{k^2+2 k} \left(\sqrt{n}-k\right)\approx \frac{6 k+1}{6}$$ A generating function $f(k)$ that removes $k$ from the output, retaining only the fractional part, ...
7
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2answers
472 views

Generalization of Bertrand's Postulate

Bertrand's postulate states that there is a prime $p$ between $n$ and $2n-2$ for $n>3$. According to Dirichlet's theorem we have that a sequaence $$a\cdot n+b$$ has infinite primes iff $a$ and $b$ ...
7
votes
3answers
769 views

Applications of cubic in number theory?

The solution of the cubic equation is known in terms of a rational function of a cube root of a square root. If we just want to know the value it's easy to approximate it using a numerical method. I ...
7
votes
1answer
976 views

Probability of cumulative dice rolls hitting a number

Is there a general formula to determine the probability of unbounded, cumulative dice rolls hitting a specified number? For Example, with a D6 and 14: 5 + 2 + 3 + 4 = 14 : success 1 + 1 + 1 + 6 + ...
6
votes
4answers
189 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
6
votes
8answers
3k views

What's the explanation for why n^2+1 is never divisible by 3?

What's the explanation for why $n^2+1$ is never divisible by $3$? There are proofs on this site, but they are either wrong or overcomplicated. It can be proved very easily by imagining 3 consecutive ...
6
votes
1answer
154 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
6
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1answer
404 views

Edmund Landau's Problems

Landau's problems are four conjectures about prime numbers which were unsolved at the time Edmund Landau presented them at the International Congress of Mathematicians in 1912. They include: ...
6
votes
1answer
322 views

a conjecture (of mine ) about primes [duplicate]

Possible Duplicate: How does $ \sum_{p \leq x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? What could I use to prove the following conjecture? $ \sum_{p \le x} p^{m} \sim ...
6
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2answers
1k views

Exploring Properties of Pascal's Triangle $\pmod 2$

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider Pascal's Triangle taken $\pmod 2$: For simplicity, we will call a finite string of 0's and 1's proper if it ...
6
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2answers
401 views

An Interesting Question about Pythagorean Triples

I have recently thought about a interesting question about Pythagorean Triples. Consider such a right-angled trapezium formed by 3 right-angled triangle. Determine does it exist integral ...
6
votes
2answers
126 views

Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primes

Is there a functon that counts the number of ways in which an even number can be expressed as a sum of two primes?
6
votes
4answers
324 views

Almost a perfect cuboid

While reading a very old book on diophantine equations, I came across this exercise: Find an infinite number of positive integer solutions of the equations $$x^2 + y^2 = u^2$$ $$y^2 + z^2 = v^2$$ ...
6
votes
2answers
567 views

Is there any famous number theory conjecture proven impossible to be find out the truth or false?

Is there any famous number theory conjecture proven undecidable? Is there any history about it? i would like to know any number theory conjecture by the types of undecidable.
6
votes
1answer
488 views

Pythagorean triplets

Respected Mathematicians, For Pythagorean triplets $(a,b,c)$, if $c$ is odd then any one of $a$ and $b$ is odd. Here $(a, b, c)$ is a Pythagorean triplet with $c^2 = a^2 + b^2$. Now, I will ...
6
votes
1answer
292 views

Can a Pratt certificate for a prime be found in polynomial time?

Can a Pratt certificate for a prime be found in polynomial time? I guess this is the same as asking whether the AKS primality test provides extra information that allows $p-1$ to be factored quickly. ...
6
votes
1answer
346 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
5
votes
2answers
78 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
5
votes
1answer
74 views

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$ where $v_2(n)$ is the largest power of $2$ dividing $n$. I think one way to solve this is to use the binomial theorem with $2005=2003+2$, but ...
5
votes
1answer
103 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
5
votes
1answer
141 views

What was Lame's proof?

In 1847, Lame gave a false proof of Fermat's Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity. The best description I've found is in the book ...
5
votes
2answers
152 views

Can an odd perfect number be divisible by $5313$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $5313$.
5
votes
1answer
103 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
5
votes
1answer
566 views

Divisibility of binomial coefficient by prime power - Kummer's theorem

Let's say we have binomial coefficient $\binom{n}{m}$. And we need to find the greatest power of prime $p$ that divides it. Usually Kummer's theorem is stated in terms of the number of carries you ...
5
votes
1answer
188 views

A conjecture about the difference between consecutive primes with respect to a prime number squared.

Conjecture If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$, so that $p_{a} < p_{a+1} < p^2_{n+1}$, then $p_{a+1} - ...
5
votes
5answers
921 views

Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.

This is also a question on my exam paper that i proved by using mathematical induction. However, my tutor tells me that it can be proved without using mathematical induction. I really want to know how ...
5
votes
3answers
117 views

How is $x^2 + x + 1$ reducible in $\mathbb{Z}_3[x]$?

I am going through my number theory notes and have got on to the bit about the ring $\mathbb{Z}_p[x]$, where $p$ is prime, and unique factorisation domains. The example I am looking at is to do with ...
5
votes
1answer
214 views

A Gauss sum like summation

I would like to calculate the following sum. Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime. The sum is $$\sum_{j=1}^n (-1)^j ...