# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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). ...
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### 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: ...
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### 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 ...
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### 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)
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### $(2^a -1)(2^b -1)=2^{2^c}+1$ 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: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
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### $\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$, 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 ...
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### 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 ...
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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 $p \equiv 5 \mod8\Rightarrow$ $p \equiv 1 \mod4\Rightarrow$ $n^{2}+m^{2}=p\equiv ... 1answer 243 views ### Adelic/Idelic method for number fields I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ... 0answers 55 views ### Confusing application of power residue reciprocity in Milne's CFT Hey I am trying to figure out the details of the proof of Theorem 5.14 (p.246) in Milne's CFT (see here). I hope somebody is familiar with this. But let me sketch the proof and what I don't understand.... 0answers 134 views ### Prime number theorem for Dedekind domains Let$\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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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. ... 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{\...
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 ...
### 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 ...