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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0
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3answers
45 views

Can $x+y=a$ and $xy=b$ be satisfied by more than one set of (x,y) when x and y are integers from 2 to 99?

Let's say we want our integer number pair (x,y) that ranges from 2 to 99 to satisfy $x+y = a$ and $xy=b$. My question is, is there $a$ and $b$, integers that have more than one pair (x,y) that ...
2
votes
4answers
182 views

Prove that $\sqrt{2 + 9n}$ is never an integer

I'm trying to show the equation $$x^2 \equiv 2 \mod 9$$ has no solutions, and I thought the best way might be to show that $\sqrt{2 + 9n}$ can never be an integer (for integer $n$). What might be a ...
0
votes
1answer
58 views

Find more counterexample mathematically or algorithmically

Problem: Find integer $N$ such that it can NOT be expressed as $N=a^2+b^2+c^2$ or $N=a^2+b^2-c^2$ where integers $0<a^2,b^2,c^2\leq N$. For $N<100000$ there should be only 17 such integers. ...
7
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1answer
202 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the $...
6
votes
1answer
698 views

How to prove that Fibonacci number is integer?

How to prove that formula for Fibonacci numbers are always integers, for all $n$: $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{...
8
votes
2answers
246 views

Erdős's exercise.

I have tried to solve an exercise I saw in "Topics in the theory of numbers" (Erdős & Suranyi) many times but failed every time I tried. Here it is: Prove that if $a_1,a_2,\cdots$ is an ...
2
votes
0answers
58 views

On the special value of Hecke L function.

For a nontrivial Hecke character $\chi:A_Q^{\times}/Q^{\times}\to S^1$, we know $L_Q(s,\chi)$ is nonzero. Is this true for number field $F$? I know is is holomorphic at $s=1$ by Artin conjecture, but ...
0
votes
2answers
63 views

How to determine if some $x$ is a generator of a subgroup of $Z^{*}_{y}$ of order $a$

Suppose we have integers $x,y$ and the prime factorization of $y-1$, and further suppose that $a$ is the largest prime factor of $y-1$ and that $y$ is prime. How do you determine if $x$ is a generator ...
1
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2answers
35 views

Ways of writing $n=2a+b$ with $a$ and $b$ are non-negative integers

For a non-negative number $n$, let $r_n$ be the number of ways of writing $n = a + 2b$, where $a$ and $b$ are non-negative integers. For example, $5 = 1 \cdot 5 + 2 \cdot 0 = 3 \cdot 1 + 1 \cdot ...
1
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3answers
3k views

Find the least nonnegative residue

Find the least nonnegative residue of $5^{18} \mod 11$. To do this I took $5^2 \equiv 3 \mod 11$. Then I did $(5^2)^5 \equiv 3^5 \mod 11$. And $3^5 \equiv 1 \mod 11$. So now I have $5^{10} \equiv 1 ...
2
votes
2answers
92 views

Number theory division proof, powers of 2

Ok, for some reason I'm getting stuck in what might be an easy question. Here's the problem: If a and b>2 are positive integers, prove that ${ 2^{a}+1 \over 2^{b} -1} $ is not an integer. My ...
4
votes
1answer
257 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
1
vote
0answers
62 views

Maximum number of consecutively selected rows.

You have a table, where the nth column repeats itself every p_n times (mod p_n). For example with n=5, you'd get a table like this, with the first column being mod 2, with the 5th column being mod 11: ...
0
votes
1answer
24 views

For which whole numbers of variable c does the following LDE have solutions in N?

For which $\mathbb{Z}$ numbers of variable c does the Linear Diophantine Equation $cx + (c + 2)y = c + 4$ have a solution in $\mathbb{N}$ ? Can please someone explains the whole process?(I know how ...
0
votes
1answer
33 views

Proving that a certain sequence is bounded from above

Let $p_1,p_2,p_3,..$ be the sequence of primes in increasing order ($p_1=2,p_2=3,...$) .Let $x_n$ be given by: $$x_n=\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_n}-\sum_{i=1}^{n-1}\sum_{j=i+1}^n\frac{...
2
votes
1answer
121 views

Showing $\sum_{n\in\mathbb{N}}\frac{\Lambda(n)}{n}=\log (x)+O(1)$ using a given result.

I'm stuck on the following problem. Use the fact that $$\sum_{\substack{p\le x \\ p\,\text{prime}}}\frac{\log p}{p}=\log (x)+O(1)$$ to show that $$\sum_{\substack{n\le x \\ n\in\mathbb{N}}}\frac{\...
0
votes
2answers
270 views

How come that $sum$ of $all$ positive integers equal a negative rational number [duplicate]

How come that $sum$ of $all$ positive integers equal a negative rational number. $$\sum_{n=1}^\infty n = \frac{-1}{12}$$ (original screenshot)
1
vote
4answers
61 views

What are all the values of $ x \equiv a^\frac{p-1}{2} \pmod p$

Suppose p is an odd prime and a $\in$ $\mathbb{Z}$ such that $ a \not\equiv 0 \pmod p$. What are all the values of $ x \equiv a^\frac{p-1}{2} \pmod p$ ? This is what I got so far: $ x^2 \equiv a^{p-...
1
vote
2answers
97 views

Is $17$ a quadratic residue for a prime $p$?

wondering if $17$ is a quadratic residue for a prime $p$? We know that $p \equiv \pm 3 \mod{8}$ but nothing else. Thanks!
2
votes
1answer
227 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if $...
2
votes
3answers
77 views

How find this positive integer numbers $m$,such $a_{n}=2a_{n-1}+a_{n-2}$,if $2^{2011}|a_{m}$

let sequence $$a_{0}=0,a_{1}=1,a_{n}=2a_{n-1}+a_{n-2},(n\ge 2)$$ Find all positive integer numbers $m$ ,such $2^{2011}|a_{m}$ My try:since $a_{0}=0,a_{1}=1$,then $$a_{n}=\dfrac{(1+\sqrt{2})^n-(1-\...
0
votes
1answer
95 views

Application of the Robins Equality

The Robin's inequality says - If the Riemann hypothesis is true then - $$\sigma(n) < e^{\gamma}n \log(\log(n))$$ holds true for all $n \in \mathbb{N}$ Now it is proved for all $5-$ free integers ....
2
votes
0answers
41 views

How prove this $\prod_{r\mod m}r\equiv 1(\mod m)$

let $m\neq 1,2,4,p^{\alpha},2p^{\alpha}$,show that $$\prod_{r\mod m}r\equiv 1(\mod m)$$ where $p$ is prime number,and $\alpha$ is postive integer numbers. maybe this problem can use Chinese remainder ...
4
votes
0answers
73 views

$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let $P_{K,1}(\...
0
votes
1answer
54 views

Is there an efficient, general formula to verify if a number is a n-sided polygonal number?

I've seen formulas to verify if a number is a triangular number, a pentagonal number, or a hexagonal number, but I haven't seen a general formula for verifying if a number is an n-sided polygonal ...
1
vote
1answer
144 views

Modulus of a very large number, How to calculate $11386^{20635} \mod 31351$?

Having issues calculating mod of a very large number. Tried to check with previous examples but was unable to understand. Please help on the follow question. How to calculate $$11386^{20635} \mod ...
3
votes
2answers
47 views

$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions

$(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers. Any solutions using parity Approach: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
0
votes
1answer
40 views

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$, where a+b=1 and $a,b,x,y>0$ real numbers. Any hints? part (a) was showing $\frac{2}{\frac{1}{x}+\frac{1}{y}}\leq \sqrt{xy}\leq \frac{x+y}{2}$. To ...
1
vote
1answer
125 views

How to prove $\frac{(a_1 a_2\cdots a_n)^2-1}{8}\equiv\sum_{i=1}^n\frac{a^2_i -1}{8}\pmod 8$

Let $a_1,a_2,\cdots,a_n$ be odd numbers, show that $$\frac{(a_{1}a_{2}\cdots a_{n})^2-1}{8}\equiv\sum_{i=1}^{n}\dfrac{a^2_{i}-1}{8} \pmod 8$$ Special cases $n=1$: It is obvious that $$\frac{a^2_{1}...
3
votes
2answers
289 views

A cubic Diophantine equation in two variables

Find all POSITIVE integer solutions to the following cubic equation: $x^3+2x+1=y^2$. Notice how the left side of the equation resembles $x^2+2x+1=(x+1)^2$. The only solutions I've been able to find ...
2
votes
0answers
80 views

$\frac{ra}{p} + \frac{rb}{p} + \frac{rc}{p} + \frac{rd}{p} = 2 $, with $p$ prime

Let $p>2$ be a prime and let $a$, $b$, $c$, $d$ be integers not divisible by $p$, such that $\{\frac{ra}{p}\} + \{\frac{rb}{p}\} + \{\frac{rc}{p}\} + \{\frac{rd}{p}\} = 2 $ for any integer $r$ not ...
15
votes
1answer
560 views

Prove that the product of some numbers between perfect squares is $2k^2$

Here's a question I've recently come up with: Prove that for every natural $x$, we can find arbitrary number of integers in the interval $[x^2,(x+1)^2]$ so that their product is in the form of $...
0
votes
1answer
466 views

The symbol $n$? Natural numbers?

This might seem like a very basic question, but it keeps bugging me. Does the symbol $n$ mean the set of natural numbers $\mathbb{N}$ shortened to $n$ to ease writing? Or is it rather the positive ...
5
votes
2answers
520 views

At least one prime between N and N-(sqrtN)

I don't know if this is an already existing conjecture, or has been proven: There is at least one prime number between $N$ and $N-\sqrt{N})$. Some examples: $N=100$ $\sqrt{N}=10$ Between and 90 and ...
2
votes
3answers
110 views

The prime game & non-prime columns.

The prime game: We define $\Delta n$ to be the number of columns in the following table: $$ \overbrace{\begin{matrix} 1 & \color{green}2 & \color{green}3 & 4 & \color{green}5 \\ 6 ...
1
vote
2answers
45 views

How can we make any integer m>11 using 3's and 5's only?? [duplicate]

Is there any general solution of this? using 2 integers, what is the minimum number formed after which we can make any number using those 2 integers? so it says 3a + 5b = m now we know 4, 7 do not ...
1
vote
1answer
51 views

$p \equiv 5 \mod8\Rightarrow p=(2x+y)^{2}+4y^{2}$

If $p \equiv 5 \mod8$ , then $p=(2x+y)^{2}+4y^{2}$,for some x and y integers. Thanks Here is my approach: I know $p \equiv 5 \mod8\Rightarrow $ $p \equiv 1 \mod4\Rightarrow $ $n^{2}+m^{2}=p\equiv ...
9
votes
1answer
243 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
2
votes
0answers
55 views

Confusing application of power residue reciprocity in Milne's CFT

Hey I am trying to figure out the details of the proof of Theorem 5.14 (p.246) in Milne's CFT (see here). I hope somebody is familiar with this. But let me sketch the proof and what I don't understand....
5
votes
0answers
134 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
2
votes
1answer
48 views

Which primepowers can divide $3^k-2$?

I tried to get a survey which primepowers $p^n$ divide $3^k-2$ for some natural k. PARI has a function znlog, but there are some issues : Instead of returning 0, if the discrete logarithm does not ...
1
vote
1answer
56 views

Proof Check: What's the minimal n that the quadratic form $10x^2-12xy+5y^2 = n$ gets?

Firstly, I noticed that by plugging in $(1,1)$ I could get $n=3$. Next, the quadratic form is positive-definite because $a=10>0$ and $b^2-4ac = 144-4*(10)*(5) = -56 <0$. This means that the ...
2
votes
1answer
79 views

Showing primes can be of the form $16x^2+y^2$

Hello everyone I am trying to solve a question that involving primes. Show that all primes that are of the form 1 more than a multiple of 8 can be written in the form $16x^2 + y^2$. I am given the ...
2
votes
1answer
124 views

Upper bound of the jacobstahl function of primorials h(n)

This is following on from my question here: Maximal gaps in prime factorizations ("wheel factorization") The solution of my problem was the jacobsthal function applied to the product of the ...
4
votes
1answer
316 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
21
votes
2answers
2k views

Are transcendental numbers computable?

Wikipedia states: "The computable numbers include many of the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental ...
0
votes
1answer
105 views

What does it mean for a “place” to divide?

A proof I'm trying to understand refers to the set of all finite places dividing an algebraic integer x. What does this mean? I can't seem to find a definition in any of the texts I've looked at. ...
2
votes
2answers
261 views

Proof of irrationality of $\dfrac{\sqrt{8}}{\sqrt{7}}$

We have to prove that $\dfrac{\sqrt{8}}{\sqrt{7}}$ is irrational(try not to use the Rational Root Theorem) At first,we prove that the expression is not an integer. $\dfrac{\sqrt{8}}{\sqrt{7}}=\sqrt{\...
0
votes
1answer
54 views

Modular arithmetic to find the mod of a large number

If $x \equiv 23 \bmod 317$ and $x \equiv 25 \bmod 331$, what is $x \bmod 104927$? What techniques are typically used to solve problems of this nature? It doesn't seem clear to me that it can be solved ...
5
votes
3answers
146 views

Does every prime $p \neq 2, 5$ divide at least one of $\{9, 99, 999, 9999, \dots\}$? [duplicate]

I was thinking of decimal expressions for fractions, and figured that a fraction of the form $\frac{1}{p}$ must be expressed as a repeating decimal if $p$ doesn't divide $100$. Thus, $\frac{p}{p}$ in ...