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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3
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3answers
272 views

Find a positive integer such that half of it is a square,a third of it is a cube,and a fifth of it is a fifth power

I found this problem in Underwood Dudley book in Linear congruence and Chineese remainder theorem ,but I can't present it as system of Linear congruence Anyone have any idea? plz help me
1
vote
0answers
11 views

Generalized Mersenne Prime Conjecture Combined $(a,b)$ pair

As read here, it is conjectured that for and pair of integers $(a,b)$ that are coprime, not perfect ${r^{\text{th}}}$ powers, and $-4ab$ is not a perfect ${4^{\text{th}}}$ power, then there should be ...
3
votes
1answer
587 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
2
votes
0answers
21 views

How to formally use Taylor expansions for $n$th derivatives and generating functions?

When deriving Catalan numbers, the generating function takes on this form: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x}) = \frac{1}{2} (1 - f(x))$$ where $f(x) = \sqrt{1-4x}$ How does one formally show ...
2
votes
2answers
60 views

Solving $28^x \equiv 2 \pmod{43}$

How do we solve $28^x \equiv 2 \pmod{43}$? I know there are not generally efficient methods for computing the discrete logarithm which are defined for an invertible $a$ modulo $q$ by $$a \equiv t^x \...
37
votes
7answers
25k views

Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
5
votes
1answer
73 views

Sum and product of rational numbers is unity

Consider the system of equations: $$\sum_{i=1}^n X_i = 1$$ $$\prod_{i=1}^n X_i = 1$$ It is reasonably simple to show that for $n\ge 4$, this system admits a rational solution $(x_1, \dots, x_n) \...
1
vote
2answers
40 views

How to formally prove whether this function is onto or not?$ K(x) = x^ 2$ where $x \ge 0$.

$K(x) = x^2.$ The domain and range of this function comprise of non-negative real numbers. If it were real numbers instead of "non-negative" real numbers, then it seems easy to prove it by ...
0
votes
3answers
30 views

Fermat's Little Theorem and Legendre symbol

I have two questions: Q1: Why is the order of $19$ modulo $29$ equal to $28$? We know by Fermat's Little Theorem that $a^{28} \equiv 1 \pmod{29}$, but why is $28$ the smallest here? Q2: Let $\left(\...
3
votes
2answers
20 views

Derivation of Taylor expansion with the $a$ term

If we have $$f(x) = \sum_{n=0}^\infty a_n x^n$$ The $k$th derivative is $$f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} x^n$$ Which also means that $$f^{(k)}(0) = k! a_k$$ Implying ...
4
votes
1answer
114 views

For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square?

Question. For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square? Clearly, $n=24$ is one such value, and I was wondering whether this is the only value for which the above holds. The ...
2
votes
2answers
103 views

Can this function be a new test for primality?

The following function returns always 0 only if a number is not prime. $$ H(x)=\prod_{i=2}^{x-1}\left\{\left[\sum_{k=1}^{x/i}(-i)\right]+x\right\} $$ what do you think? Bye!
1
vote
2answers
49 views

Intuitive derivation of Taylor expansion?

I was looking up the derivations of Catalan numbers, and one derivation (probably the most famous) involves generating functions that leads to: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x})$$ And then this ...
3
votes
1answer
26 views

Lang-Nishimura theorem still carries through or fails when assumptions are dropped?

Theorem (Lang-Nishimura). Let $X \,-\!\!\rightarrow Y$ be a rational map between $k$-varieties, where $Y$ is proper. If $X$ has a smooth $k$-point, then $Y$ has a $k$-point. Does the theorem of Lang-...
4
votes
7answers
90 views

Prove that $p^2 - 4qr$ ($p,q,r$ odd natural numbers) is never a perfect square

The givens for the question: $p, q, r$ are odd natural numbers. We need to prove that $p^2 - 4qr$ is never a perfect square. Inspecting a few examples it seems to be true, but I have no idea where to ...
8
votes
2answers
945 views

How to find the integral closure of $\mathbb{Z}_{(3)}$ in the field $\mathbb{Q}(\sqrt{-5})$?

Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by $$ \mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}. $$ That is, $\...
2
votes
2answers
29 views

Legendre symbol simplification

I saw a simplification using the Legendre symbol which said $$\left(\dfrac{19}{29}\right) = \left(\dfrac{10}{19}\right) = \left(\dfrac{2}{19}\right) \cdot \left(\dfrac{5}{19}\right) = -1.$$ My ...
1
vote
0answers
30 views

Is this a correct definition for $p$-adic norm?

The definition of $p$-adic norm in most textbooks and here is not easy for me to understand and especially to implement in practice, but here it is the way I reworded it: The norm of a $p$-adic ...
2
votes
1answer
39 views

Can the Von-Mangoldt function and the Chebyshev function be defined for entire complex plane?

Can the von-Mangoldt function and the Chebyshev function be defined for the entire complex plane ? I assume so, but I had not seen the definition. Can anyone provide some links for this? Thank you.
1
vote
1answer
55 views
0
votes
1answer
53 views

Solve $28^x = 19^y+87^z$ [duplicate]

Solve the equation $28^x = 19^y+87^z$, where $x,y,z$ are integers. This is related to Beal's conjecture (and it turns out there are no integer solutions to the equation), but I am wondering how to ...
5
votes
3answers
82 views

Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
1
vote
2answers
33 views

Testing prime numbers with modified Fermat's Little Theorem

Is there a number $n$ such that: $6n-1$ is prime There exists a positive integer $r<3n-1$ such that $4^{r}\equiv1\pmod{6n-1}$
4
votes
2answers
41 views

Two questions on the Gaussian integers [duplicate]

I have two questions on the Gaussian integers. Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$? Conversely, does any element in $\mathbb{Q}(i)$ ...
0
votes
1answer
22 views

Using the prime number theorem to find a continuous function mapping primes?

The prime number theorem gives an increasingly (proportionally) accurate approximation for the number of primes below $x$. Can we use this to find an equivalently accurate approximation which maps the ...
2
votes
0answers
32 views
-1
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0answers
28 views

KVSIMO 2001 Number theory problem [on hold]

If $a,b$ are integers, $ a^3 +b^3 +1 = a^2b^2$ and $a\geq b$, find $a+b$.
11
votes
1answer
196 views

Are there infinitely many pairs of primes where each divides one more than the square of the other?

I have the following question on number theory that is eating my head. Are there infinitely many primes $p,q$ such that $p | (q^2 + 1)$ and $q | (p^2 + 1)$? I can see $13,5$ and $2,5$ has the ...
12
votes
1answer
96 views
+100

Are there infinitely many primes $n$ such that $\mathbb{Z}_n^*$ is generated by $\{ -1,2 \}$?

Let $n$ a prime, and let $\mathbb{Z}_n$ denote the integers modulo $n$. Let $\mathbb{Z}^*_n$ denote the multiplicative group of $\mathbb{Z}_n$ Are there infinitely many $n$ such that $\mathbb{Z}^*_n$ ...
0
votes
0answers
6 views

If I hack a maximal sequence into another sequence, is it maximal and can I derive the primitive for it?

Given a maximal sequence generated by an LFSR - I would like to perform the following post-processing on the sequence: If the number is even - discard it. Remove the bottom bit of the number (must ...
0
votes
2answers
93 views

Primes and squares

Which is the largest n such that, given $S=\{1, 2, 3, ..., 2n\}$, we can pair off the elements of S into n pairs so that each pair sums to a different prime or square?
1
vote
2answers
69 views

Solve the equation $28^x = 19^y+87^z$ in integers

Solve the equation $28^x = 19^y+87^z$, where $x,y,z$ are integers. I am confused how to find solutions for all integers and not just positive ones. How should we do that?
0
votes
0answers
24 views

on the least primitive root of a prime

There is an article in this link. I am trying to understand it but some parts seem unclear to me. For example in part 3 I don't know how the following was derived: $$ \left ( 2^{m}-1 \right )p^{\frac{...
2
votes
1answer
97 views

Is $\sqrt2+\pi$ irrational?

From this, as a layman I wonder if the same goes for $\sqrt2+\pi$? How about $\pi+\log2$?
3
votes
3answers
180 views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
1
vote
1answer
25 views

Lemma Using Proth's Theorem ${(a, a+1)}^{(p-1)/2}$ $=$ $-1$ $\pmod p$ and $p$ is composite?

Proth's Theorem: Let $p=k*2^n + 1$ where $k$ $<$ $2^n$. If there is any integer $a$ such that $a^{(p-1)/2}$ $=$ $-1$ $\pmod p$, then $p$ is prime. Can anyone please find a counterexample (...
3
votes
0answers
112 views

conjectured arithmetic properties of some continued fraction

Given the continued fraction found in this post,bearing a striking resemblance to the one in this post $$G(q)=\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q(1-q^2)^2}{1-q^5+\cfrac{q(1-q^3)^2}{1-q^7+\...
0
votes
1answer
45 views

Combinatorics-Number Theory Problem

A positive integer is written on each vertex of a pentagon, a different one on each vertex. On each side is written the $lcm$ of the numbers of the vertices that form that side. If $n$ is written on ...
2
votes
1answer
33 views

Geometric Intuition of Group Structure on Elliptic Curve

I am reading Number Theory 1: Fermat's Dream by Kato. In Chapter 1 he defines the group structure on a general elliptic curve $$y^{2} = ax^{3} + bx^{2} + cx + d$$ (where $a \neq 0$, and the cubic ...
6
votes
1answer
76 views

Prime Factorization Question involving Product of Consecutive Terms

I came across this question while doing some research at an REU this summer. It was supposed to be just a small part of a larger proof, but we've been stumped on it for a while. I don't have much of a ...
1
vote
1answer
72 views

Show that $\alpha$ is irrational

Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1,1987,1987^2,\ldots$ to the right of the decimal point. Show that $\alpha$ is irrational. In order ...
2
votes
1answer
47 views

Number theory problem, fractions and gcd, please help!!!

The problem says "if a,b are positive integers such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer then show that $\sqrt{a+b}\ge$ gcd(a,b)" Adding $\frac{2ab}{ab}$ to $\frac{a+1}{b}+\frac{b+1}{a}$ ...
0
votes
1answer
40 views

How accurate is the approximation of the number of rough numbers?

A number is called a $y$-rough number, if it has no prime divisor below $y$. The number of rough numbers in an interval, lets say, $[10^{99},10^{100}]$ is approximately the length of the interval ...
1
vote
4answers
40 views

Infinite set of positive integers such that the greatest common divisor of any two distinct numbers in $B$ is $p$

Let $A$ be an infinite set of positive integers such that every $n \in A$ is the product of at most $k$ prime numbers where $k$ is a positive integer. Prove that there are an infinite set $B \subset A$...
0
votes
2answers
75 views

Primality testing though trial division.

I am having difficulty to understand this statement mentioned here: Remember that any composite integer n is build out of two or more primes n = P * P … P is largest when n has exactly two ...
4
votes
1answer
149 views

Confused by a proof about harmonic numbers

I've been puzzled by a step in D'Aurizio's proof concerning the finiteness of a certain subset $J_p$ of $\mathbf{N}$: $$J_p = \{n : p \text{ divides the numerator of } H_n\}.$$ His paper is here: ...
-1
votes
2answers
52 views

Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.

My approach: let $x=3 \lambda, y=4 \lambda,z=5 \lambda $ where $\lambda \in \mathbb Z$ ,then $x^2+y^2=z^2 \implies $ $(3 \lambda)^2+(4 \lambda)^2=(5 \lambda)^2$ Since $xyz=60 \lambda^3$ & $60 \...
1
vote
0answers
53 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
2
votes
4answers
258 views

The Frobenius Coin Problem

I am asked to prove that: For integers $n, x,y > 0$, where $x,y$ are relatively prime, every $n \ge (x-1) (y-1)$ can be expressed as $xa + yb$, for $a,b \ge0$. How should I approach ...
2
votes
3answers
63 views

Integer solutions to $x^3+y^3+z^3 = x+y+z = 8$

Find all integers $x,y,z$ that satisfy $$x^3+y^3+z^3 = x+y+z = 8$$ Let $a = y+z, b = x+z, c = x+y$. Then $8 = x^3+y^3+z^3 = (x+y+z)^3-3abc$ and therefore $abc = 168$ and $a+b+c = 16$. Then do I just ...