Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Absolutely and uniformly convergence and entire functions

I'm studying transcendental functions, and I'm reading an article about it. On the proof of one of the lemmas, the author gets that: $$\sum_{n=0}^{\infty}|a_n||P_n(z)|\leq e^{\max\{|z|,1\}},$$ where ...
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1answer
25 views

Find extra work done by Bob

Alice has challenegd Bob game of N puzzle.N puzzle is played on N*N grid with each cell containing distinct numbered tile from 1 to N*N-1 Except one which is empty cell and represented as 0. Move ...
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0answers
23 views

variation of the Euler $\phi$ function?

Let $n \leq m$ be positive integers. Is there a function or expression giving the cardinality of the set $\{r \in \mathbb{Z}^+| 1 \leq r \leq m, \gcd(r,n) = 1 \}$? If $n = m$, it's just $\phi(n)$.
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3answers
28 views

If algebraic $a$ has degree $n$, so does $-a$

I feel like the best way to move forward is to use a contradiction proof. Since $a$ is algebraic, and is of degree $n$, it has a minimal polynomial of degree $n$, so we can write ...
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2answers
60 views

how to find taxicab numbers but for squares?

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205. $$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 ...
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2answers
53 views

minimal polynomial given an algebraic number

I am trying to find the minimal polynomial for the algebraic number $1+\sqrt{2}+\sqrt{3}$. My original thought was just let $\alpha=1+\sqrt{2}+\sqrt{3}$. The method I use though seems very ...
6
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1answer
115 views

Can the product of $n$ factorials be $n$ factorial?

Are there any solutions to the equation $a_1!\cdot a_2!\cdots a_n!=n!$ with all variables being integers greater than or equal to $2$?
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1answer
82 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
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0answers
79 views

“Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
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2answers
65 views

Proving the divisibility of large numbers without making large calculations [duplicate]

How would you you show that $2^{32}+1$ is divisible by $641$ without making large calculations?
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2answers
54 views

Difficult generating function

Prove that the coefficient of $x^i$ for $i=1,2,3,...$ in the expansion of $\prod_{n=0}^{\infty} (1+x^{3^n})(1+x^{4^n})(1+x^{5^n})(1+x^{6^n})$ is greater than $0$.
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2answers
31 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
3
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1answer
80 views

Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
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0answers
30 views

How to have this equation $s^2-2(p+q+r+2pqr)s+(p^2+q^2+r^2-2(pq+qr+rp)-4)=0$?

Old Question: For $x,y,z\in N^{+}$, if such $(xy+1)(yz+1)(zx+1)$ is a perfect square ,show that $$(xy+1),(yz+1),(xz+1)$$ are all perfect square . and I konw this PDF have solution, ...
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3answers
66 views

Prove that there exists $s$ such that $s(ab-1)^n +1$ is composite

I find this interesting question in a number theory book. Given two positive integers $a, b$ such that $a>1, b>1, \gcd(a, b)=1$. Prove that there exists a positive integer $s$ such that ...
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2answers
39 views

Proving Fibonacci inequality

I didn't see a question regarding this particular inequality, but I think that I have shown by induction that, for $n>1$. I am hoping someone can verify this proof. ...
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0answers
27 views

general local to global principle

Consider the Diophantine equation $f(x)=0$, where x is a vector of integers and $f: \mathbb Z^n \rightarrow \mathbb Z$ is a polynomial function. Is the following statement true? The structure of the ...
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2answers
186 views

Solutions to the Mordell Equation modulo $p$

It is well known that the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions, but has solutions modulo $n$ for all $n$. One proof of this involves using the Weil Bound to show that $x^2 = ...
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1answer
32 views

Find the number of positive integer $a \leq n$ such that $(a,n) = (a+1,n) = 1)

For every positive integer $n$, let $$A_n = \{a \in \mathbb{N} \mid 1 \leq a \leq n \mid gcd(a,n) = gcd(a+1, n) = 1\}$$ Evaluate $\mid A_n\mid$ Assume that $n$ has the factorization ...
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0answers
43 views

How prove this $3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$

Question: show that: $$3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$$ My idea: since I have prove $$5^{2^n}-1\equiv 0\pmod {2^{n+2}}$$ ...
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1answer
50 views

fastest algorithm for prime factorization [on hold]

I need the fastest algorithm to factorize the given number $N$ as a product of primes. $$N=p_1^{e_1}p_2{e_2}\ldots p_n^{e_n}$$ where $p_1, p_2,\ldots ,p_n$ are primes and $e_1,e_2,\ldots, e_n$ are ...
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2answers
43 views

Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
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0answers
24 views

Mobius function verification

I am looking to verify my answer to the question $$F(n)=\sum_{d|n}{\mu(d)\sigma(d)}=(-1)^{\omega(n)}\prod_{j=1}^{\omega(n)}{p_j}$$ Where $\mu$ is the Mobius function, $\sigma$ is the sum of divisors ...
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2answers
40 views

Diophantine solution to a fraction

How can we find solutions to the following equation: $$ y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
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0answers
38 views

How does this method work? [on hold]

Let $n=16$ for an example: step 1: get set of prims from $1$ to $\sqrt{2n}: \{2, 3, 5\}$, step 2: get set of $n \mod 2, n \mod 3, n \mod 5: \{0, 1, 1\}$, setp 3: from $0$ to $n-3$, ...
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1answer
25 views

Sizes of Blocks of Consecutive Integers Divisible by at Least One Prime Less than or Equal to $r$.

Let $f(r)$ be the largest integer such that there exists a block of $f(r)$ consecutive integers each divisible by some prime that is less than or equal to $r$. For example, $f(2)=1$ because it is ...
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1answer
35 views

regarding pseudo-prime numbers.

If $W$ is an odd composite number and $-1+2^{W-1}$ is divisible by $W$ yet not by $W^2$, then $W^2$ does not divide $-1+ 2^{W(W-1)}$. Is this true? (forgive my use of symbols,I have no good math ...
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4answers
117 views

What is the remainder when the below number is divided by $100$?

What is the remainder when the below number is divided by $100$? $$ 1^{1} + ...
2
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2answers
72 views

Fibonacci number ending with given sequence of digits

Related to this question: For any given sequence of digits, does a Fibonacci number exist ending with such sequence? If not, it would be nice to find the smallest counterexample. (in other ...
2
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0answers
26 views

Polynomial Density Theorem

So I was consider Lagrange's 4-square theorem and came up with this generalization: Given a polynomial with rational coefficients $$P(x) = a_0 + a_1x + ... + a_nx^n$$ Determine if there exists ...
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2answers
103 views

Finding all possible pairs of integers $(a,b)$ such that $a^b=n$.

Given a large integer $n$ (could be as large as $10^{18}$), how can I find all possible pairs of integers $(a,b)$ such that $$a^b=n.$$ A fast algorithm is preferable. The question How to quickly ...
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0answers
47 views

How do ramanujan sums and the sum of 2 squares relate?

I read on the Wikipedia page for the ramanujan sum a formula for the sum of 2 squares in terms of the ramanujan sums. But I did not get it. http://en.wikipedia.org/wiki/Ramanujan_sum How do ...
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107 views

Fermat's Last Theorem 1 Page Proof [on hold]

Fermat’s Last Theorem This theorem basically states that $A^n + B^n \neq C^n$, $n \gt 2$ if $A, B, C$ and $n$ are all positive integers. This inequation can be rewritten as $C^n – B^n \neq A^n$. ...
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1answer
52 views

Contracted ideals in number fields

I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement: Suppose for ...
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22 views

Fermat pseudoprimes p to base 2 (AKA Sarrus or Poulet numbers) with special properties

Are there any known Fermat pseudoprimes $p\;$ to base $2\;$ (Sarrus or Poulet numbers) with the properties $q = (p-1)/2\;$ is prime and $p \equiv 0 \pmod 3?$ I was not able to find any example up to ...
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1answer
30 views

How find the minimum of the value $n$ such$105\mid \left(9^{p^2}-29^p+n\right)$

Find the minimum of the value $n$,such for any prime number $p>3$,have $$105\mid\left(9^{p^2}-29^p+n\right)$$ My since $$105=5\times 3\times 7$$ so ...
3
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0answers
51 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
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0answers
49 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
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3answers
246 views

Is there a polynomial equation for $f(n) = n!$ and if so what is it?

And I am not necessarily talking about $f(n) = n(n-1)(n-2)...(3)(2)(1)$ in its factored form; Well it could be that but then I would like a general way of expansion. Thanks in advance!
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2answers
1k views

Proof that 26 is the one and only number between square and cube

$x^2 + 1 = z = Y^3 - 1$ Why $z = 26 $ and only $26$ ? Is there an elementary proof of that ?
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24 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
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1answer
18 views

A formula occurring in Dirichlet's proof of the infinity of primes in an AP.

While studying Dirichlet's proof of an infinity of primes in any AP with first term and common difference coprime, the formula below involving the gamma function was quoted as being well known. ...
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88 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
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1answer
32 views

Solution to equation in surd

$\sqrt{a+b\sqrt{c}}=\sqrt{x}+\sqrt{y}$ where $a, b, c\in\mathbb{Z}^+$ and x, y $\in \mathbb{Q} $ Please help show how to disprove or prove. Thanks a lot
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1answer
17 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
15
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3answers
317 views

Equality of Sums

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
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1answer
38 views

Find $x$ such that $o_1^x-o_2^x \neq 2(o_3^x-o_4^x)$ where $o_i$ is an odd number, $o_1>o_2$, $o_3>o_4$ and $x$ is a positive integer

A few hours ago I asked this question. This problem came up while working on a graph labeling problem. I already have a exponential algorithm working. But I want to further reduce the complexity. ...
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2answers
40 views

Number theory proof regarding congruences and common divisors

Anyone know how to prove the following statement? If $ a=b $ (mod m) then the common divisors of a,m are the same as those of b,m
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22 views

Given an integer n find smallest integer i such that σ(i)=n. Smallest Inverse Sum of Divisors

Hi All I need some help I am trying to solve this problem which involves computation of sum of divisors and its inverse. In other words Given an integer n find smallest integer i such that σ(i)=n ...
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1answer
39 views

Number theory proof regarding norms

How would you prove that if $x$ is a prime in $ℤ[i] \Longleftrightarrow$ $N(x)$ is a prime in $ℤ$ N(x) represents the norm of x.