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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5
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0answers
103 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
9
votes
1answer
132 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
1
vote
1answer
72 views

How do you find the smallest legitimate encryption exponent when you are only give a p and q value in a given range?

I have been given this as an assignment question but I'm not sure approach it. EDIT: Sorry I should have added more details. It is a cryptosystem using the RSA scheme. p and q are both old prime ...
6
votes
1answer
60 views

$p+q\neq 1+pq$ for distinct odd primes $p$ and $q$

I'm trying to show that that $\sqrt p + \sqrt q$ cannot be written as a linear combination of $1$ and $\sqrt{pq}$ with rational coefficients, and I have boiled it down to showing that $p+q \neq 1+pq$ ...
1
vote
2answers
40 views

Finite sums of integers and similar problems: book request

I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
3
votes
1answer
49 views

What does this use of little-o notation mean?

I am currently going through the proof of Prime Number Theorem, as given in Hardy and Wright, and in it they define the following constant: $$\alpha = \limsup_\limits{x \to \infty} \left|V(x) ...
1
vote
2answers
78 views

Explain the proof of irrationality of $\sqrt{2}$

How does this proof show the irrationality of $\sqrt{2}$ ? I am new to proofs and don't really understand the logic used here.
4
votes
2answers
54 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
0
votes
1answer
52 views

Expression of theorem by $p\Rightarrow q$ [closed]

$\sqrt 2$ is irrational. Is it true that i express this theorem in this way?: If $\sqrt 2$ is a real number, then it is irrational. Is there any better way to express this theorem by $p\Rightarrow q$? ...
1
vote
1answer
27 views

Minimum number of elements required to make sum?

I want the minimum number of elements to get a required sum. For Example:Sum=4 N=3 {1,2,3} is enough since (1+3 = 4)
2
votes
1answer
30 views

The diophantine problem for $R[T]$ is solvable iff the diophantine problem for $R$ is solvable

One part of the paper that I am reading is the following: Let $R$ be a commutative ring with unity and let $R'$ be a subring of $R$. We say that the diophantine problem for $R$ with coefficients ...
7
votes
1answer
54 views

Two definitions of ramification groups, why are they equivalent?

Let $L|K$ be a finite galois extension and suppose that $v_k$ is a discrete normalized (non-archimedean) valuation of $K$ with positive residue field characteristic $p$, and that $v_K$ admits a unique ...
1
vote
2answers
48 views

Reflection formula for Hurwitz Zeta function?

In doing some calculus with Mathematica today, I found that $$\zeta\left(3,\frac{1}{4}\right) - \zeta\left(3,\frac{3}{4}\right) = 2\pi^3$$ by numerically evaluating both sides. Here, $\zeta(x,y)$ ...
3
votes
3answers
85 views

When is the difference of two consecutive positive cubes a perfect square?

Are there only finitely many solutions in positive integers $m,n$ to the equation $$(m+1)^3-m^3=n^2\; ? $$
2
votes
3answers
84 views

Prove $5^{31}13^{25}$ can be represent as $ a^2+b^2;a,b \in \mathbb{Z} $

Prove $5^{31}13^{25}$ can be represent as $ a^2+b^2;a,b \in \mathbb{Z} $ I read about complex numbers. Authors represent formula: $$({a_1}^2 + {b_1}^2)({a_2}^2 + {b_2}^2)\, = \,{({a_1}{a_2} - ...
0
votes
1answer
29 views

Finding Numbers where modulo is k

I have given a number $A$ where $1\le A\le 10^6$ and a number $K$. I have to find the all the numbers between $1$ to $A$ where $A\%i=k$ and $i$ is $1\le i\le A$. Is there any better solution than ...
1
vote
1answer
17 views

If $a$, $b$, $z$ and $y$ are integers such that $\gcd(a,b)=1$ and $x^a=y^b$, show that $x=n^b$ and $y=n^a$ for some integer $n>1$

I first noted that the set of primes dividing $x$ will be the same as the set of primes dividing $y$. Then i assumed $x=p^l$ and $y=p^m$ (where $p$ is any prime dividing $x$ and $y$ and $p^l$ is the ...
27
votes
1answer
325 views

Uses of “Collatz induction”?

The Collatz conjecture is equivalent to the following "induction principle": If $P(0) \land P(1) \land (\forall{x} P(3 \cdot x + 2) \implies P(2 \cdot x + 1)) \land (\forall x P(x) \implies P(2 \cdot ...
3
votes
3answers
68 views

Prove that if $a,b,c$ are positive real numbers

Prove that if $a,b,c$ are positive real numbers, then the least possible value of $$6a^{3}+9b^{3}+32c^{3}+\frac{1}{4abc}$$ is 6. For which values $a,b$ and $c$ is equality attained ? I know how ...
0
votes
0answers
28 views

Third separated fano plane

Let's suppose that we have a set named $A = \{1,2,3,4,5,6,7\}$. Now if we write whole $3$ membered subsets of it, we want to make fano plane of them. Then we removes that subsets from list and make ...
0
votes
1answer
22 views

Does coprimality also extend to inverses?

For example, consider the congruence, for positive $x,y$ $$5^{y-10} = x5^{y-10} \bmod 64$$ Is it safe to divide both sides by $5^{y-10}$? Clearly if $y-10$ is non-negative, it will be coprime to 64 ...
3
votes
1answer
29 views

Proof of elements in 4 digits palindrom

Can you prove that there are exactly 90 elements in the set of numbers having 4 digits which are palindromes? This is not a tricky question. I am just trying to understand the concept of proofs ...
0
votes
0answers
33 views

Truth of second Nyonyon theorem

This is the second of 5 Nyonyon theorems. Let s = ab be a semiprime number, and let R(X) be the reversal of x, and all Dots here means concatenation, then second Nyonyon theorem states that if 11 ...
2
votes
1answer
99 views

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one ...
1
vote
1answer
28 views

Asymptote of $\sum_{j=1}^{n}(-1)^{j} \phi (j)$

How do you compute the asymptotic behavior, not sure if I'm wording this correctly, of $\sum_{j=1}^{n}(-1)^{j} \phi (j)$ where $\phi$ is the euler totient function, as in ...
4
votes
1answer
46 views

Gauss sums and module endomorphisms

Let $p$ be an odd prime and $n \in \mathbb{N}$. Let $a,b,c$ be arbitrary integers such that $ab \neq 0$. We write $p^{\alpha}A = a$ and $p^{\beta}B = B$ for some $\alpha, \beta \in \mathbb{N}_0$ and ...
7
votes
2answers
111 views

First-order definition of nonnegative in integers

Given the structure $(\mathbb Z,+,-,\times,0,1)$, what's the easiest way to write "$x\ge0$" in that structure? I know that this works: $$\exists a\exists b\exists c\exists d,a^2+b^2+c^2+d^2=x$$ ...
0
votes
0answers
23 views

Relation between number coding in shortlex order for different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
0
votes
1answer
46 views

Number theory;Mit maths for computer Science

I am reading the book of MIT MATHS FOR COMPUTER SCIENCE.BUT GOT CONFUSED HERE: The rule of thumb derived from the Prime Number Theorem says that among 10-digit numbers, about $1$ in $\ln ...
1
vote
1answer
38 views

Counting non-decreasing integer sequences with a condition

I am having difficulty framing this properly. How many non-decreasing integer sequences are there of length $n$, where each element is bound between $1$ and $m$ inclusive, such that the longest ...
6
votes
1answer
78 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
1
vote
0answers
36 views

How do we identify if a number can be represented as the sum of squares of$ 3$ integers? [duplicate]

How do we identify if a number can be represented as the sum of squares of $3$ integers? For eg: $434 = 11^2 + 12^2 + 13^2$ but $432$ is not?
1
vote
0answers
27 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...
0
votes
3answers
67 views

if $\left\lfloor\dfrac{2^n}{n}\right\rfloor $is a power of $2$,then $n$ is also a power of $2$?

If $n$ is a positive integer such that $n\ge 4$ and $\left\lfloor\dfrac{2^n}{n}\right\rfloor$ is a power of $2$; then $n$ is also a power of $2$. Example $n=4,n=8,16,\cdots,2^k$, then ...
1
vote
1answer
49 views

How one can solves an equation of the form: $ap_{n}+bn=c$

My question is: How can one solve an equation of the form: $$ap_{n}+bn=c$$ where $p_{n}$ is the $n^{th}$ prime number, $a,b$ and $c$ are integers.
3
votes
1answer
169 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
2
votes
3answers
58 views

Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric progression.

Let $a,b,c,d$ be real numbers not all $0$, and let $f(x,y,z)$ be the polynomial in three variables defined by $$f(x,y,z) = axyz + b(xy + yz + zx) + c(x + y + z) + d.$$ Prove that $f(x,y,z)$ ...
2
votes
1answer
29 views

An abelian number field is either totally real or CM-field

The wikipedia article of totally real number fields says: The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally ...
1
vote
0answers
13 views

Non-CM totally imaginary number fields

Is there a name for the totally imaginary number fields that are not CM-fields? Any important subclass of number fields with that property, or perhaps a reference where those field are studied in ...
1
vote
4answers
147 views

Sum of non-real roots of equation?

What is the sum of all non-real, complex roots of this equation - $$x^5 = 1024$$ Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. ...
3
votes
1answer
49 views

The largest subset of a finite cartesian product in which distinct elements differ in at least 2 components

Let $A_1,\ldots,A_n$ be finite sets of sizes $a_1,\ldots,a_n$. What is the largest possible size of a subset $S\subset\bigotimes A_k$ such that if $(d_1,\ldots,d_n),(e_1,\ldots,e_n)\in S$, then ...
2
votes
3answers
134 views

Formula for alternating sequences

I am looking for a general formula for alternating sequences. I know that the formula $f(x)=(-1)^x$ gives the sequence $1,-1,1,-1,...$ but I want more a general formula; for example the function ...
3
votes
1answer
58 views

How many different tables can we generate?

We may permute the rows and the columns of the table below. How many different tables can we generate? A hard problem. It's for discussion, thanks!
1
vote
0answers
38 views

Can this combinatoric sum be simplified?

Base cases: $F(n,k,d) = 0$ if $d=0$ and $n>0$ $F(n,k,d) = 1$ if $n=0$ Expression: $$F(n,k,d) = \sum_{s=0}^{\min(k,n)}\binom{n}{s}F(n-s,k,d-1)$$ I am trying to compute the value of $F(n,k,10)$ ...
0
votes
5answers
186 views

best approximation of $\sqrt{2}$

The approximation \begin{align} \sqrt{2} &\approx \frac{1}{8} \operatorname{csch}\left(\frac{3\pi}{2}\right) \operatorname{sech}^3(\pi) \, \left[2+3 \, ...
2
votes
2answers
104 views

How does WolframAlpha solve quintic equations (5th degree polynomials)?

There exists no general formula for quintic equations. Then how does WolframAlpha solve quintic or higher degree polynomial. Is there a sure way to get values of roots of quintic or higher degree ...
2
votes
1answer
28 views

Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$

Let $\ell^\star$ be the upper logarithmic density on the set of positive integers, namely $$ \forall X\subseteq \mathbf{N}^+, \,\, \ell^\star(X)=\limsup_n \frac{1}{\ln n}\sum_{x \in X\cap ...
1
vote
0answers
19 views

Periodic tribonacci-like sequence

How to prove that if $a_n =[(t_{n-3} + 2t_{n-2} + t_{n-1}) a_{1} + (t_{n-3} + t_{n-2} + 2t_{n-1})] \quad (\text{mod}10)$ and $a_{1}, a_{2}, a_{3}$ are consecutive numbers and $t_{1}=0, t_{2}=1$ and ...
2
votes
1answer
33 views

Completion and algebraic closure commutable

The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of ...
11
votes
4answers
227 views

Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?

In another question, posted here by jordan, we are asked whether it is possible to cover the numbers $\{1,2,\ldots,100\}$ with $20$ geometric sequences of real numbers. Naturally, we would like to ...