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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2answers
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How to prove $~\sqrt{3}\sqrt{4a^{3}-1}~$ isn't an integer?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. If $~(c−b)=1~$ then $~c^3-b^3=3c^2-3c+1=a^3,~$ from Wolframalpha get: $$ c = \dfrac{3- \sqrt{3}\sqrt{4a^{3}-1}}{6} \\ c = ...
3
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1answer
30 views

Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
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6answers
54 views

Find 2 square numbers with certain distance

Is there a way to find 2 sqare numbers with a certain distance without trying every square number? Example: $$ a^2 + 204 = b^2 $$
1
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0answers
17 views

Count arrangment such that each person wear different tshirt

Few friends are going to a party. Each person has his own collection of T-Shirts. There are 100 different kind of T-Shirts. Each T-Shirt has a unique id between 1 and 100. No person has two T-Shirts ...
1
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1answer
38 views

Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?

As we know that $f(x)=x^2+1\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$, does there exist a cubic polynomial $f(x)=ax^3+bx^2+cx+d~(a,b,c,d \in\mathbb Z,a\neq 0) $ such that ...
0
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0answers
29 views

What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the ...
0
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1answer
18 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
2
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1answer
27 views

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
0
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3answers
164 views

1729, and related questions

I just read this paragraph: (written by G. H. Hardy, on Ramanujan) I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the ...
2
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1answer
51 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 ...
3
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1answer
47 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
5
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1answer
44 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
3
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1answer
43 views

Calculate 'interference' of number patterns

I have 2 numerical series like this: $$ 144 + 25 + 27 + 29 + 31 + \cdots $$ $$ 133 + 3 + 5 + 7 +9 +11+13+\cdots $$ Is there a efficient way to find the common sum of these patterns? solution for ...
2
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1answer
39 views

Unique Products on a Times Table

I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed ...
0
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0answers
12 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
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1answer
74 views

Squeezing $\pi(x)$ out of $\psi(x)$

Can $\pi(x)$ be written in terms of $\psi(x)$? I can only seem to approximate it: $$ ...
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0answers
35 views

Sparsity of Linear Diophantine Equations

If you are looking for integer solutions to the system. $$Ax=b$$ where $A$ is an integer matrix and $b$ is integer vector, then you can construct the solution space integer matrix $B$ and integer ...
1
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2answers
41 views

Can't understand source of constant for prime counting function:

Consider the prime counting function $$ \pi(x) = \ the \ number \ of \ primes \ less \ than \ or \ equal \ to \ x$$ It is well known due to the sieve eratosthenes that given an integer $n$ and the ...
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4answers
140 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
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1answer
59 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
2
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2answers
66 views

Diophantine equation: $n^p+3^p=k^2$

Find all solutions to the Diophantine equation $n^p+3^p=k^2$, where $p\in \mathbb{P}$ and $n,k$ positive integers. I have tried everything, from mods to bounding to LTE; nothing seems to work on ...
3
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0answers
34 views

Having trouble with binary quadratic forms.

A quadratic form represents an integer $n$ if there exist $x,y\in \mathbb{Z}$ such that $f(x,y)=n$. It is proper if $\gcd{(x,y)}=1$. It is said that if $f(x,y)=n$ and $\gcd{(x,y)}=g$, then $g^2|n$. ...
0
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1answer
20 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
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0answers
43 views

Infinitely many prime numbers 6n-1 [duplicate]

Prove that there are infinitely many prime numbers of the form $6n-1$. I proved that there are infinitely many prime numbers but I couldn't bring it in the form given in the question. While proving ...
2
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0answers
36 views

What's the proof for the #integers less than $n$ that can be expressed as the sum of two squares is $\frac n{\sqrt{\log n}}$?

This result is used in the Erdos' Distance problem, in the Landau-Ramanujan constant, but I can't find a proof anywhere. http://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem ...
2
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1answer
43 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...
1
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1answer
25 views

Expanding Base 2B representation of an integer

Consider an integer$L$ written in Base 2B which digits $$a_n a_{n-1} a_{n-2} ... a_1 B$$ Where $a_i$ are arbitrary constants such that $9 \le a_i < 2B$. I am attempting to prove that the square ...
0
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0answers
24 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bares some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
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2answers
243 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
3
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1answer
30 views

Non unique factorization domains with prime factorizations with differing number of primes

As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
3
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0answers
35 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
0
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1answer
87 views

Number theory proof regarding primes and the number of digits of the prime [duplicate]

How would you prove that if given a prime each of whose (decimal) digits is equal to $1$, then the number of its digits is a prime. (It is not known if there exists infinitely many such prime)
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3answers
32 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
2
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2answers
59 views

Is 7 prime or irreducible or something else in $\mathbb{Z}_{21}$

I thought I understood prime numbers pretty well, but now I'm told about this thing called irreducible, that sometimes numbers are irreducible but not prime (like 3 in $\mathbb{Z}[\sqrt{-5}]$) and ...
3
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3answers
110 views

Does $x^2+x+1 \equiv 0 \pmod {997}$ have solutions? Why or why not?

I'm have difficulty solving this problem in my textbook. Does $x^2+x+1 \equiv 0\pmod{997}$ have solutions? Why or why not? I guess the first step would be $$ \begin{array}{l} (2x+1)^2 \equiv ...
1
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0answers
21 views

Asymptotic behavior of sums of consecutive powers (bivariate)

Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum: $$\sum_{k=1}^{n} k^d,$$ where $n$ and $d$ are large integers? I am especially interested in a lower bound of ...
0
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2answers
19 views

Scan through all integers within a range by incrementing constant amount

Given a natural number $k$, what is a way to find out all natural number $i$ such that, when we start with $n = 0$ and keep adding $i$ to $n$, the value $n \mod k$ traverses through all numbers ...
1
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1answer
45 views

Equivalent definitions of a lattice in a real vector space of finite dimension

I'm currently trying to work my way through chapter seven of Serre's book "A Course in Arithmetic" with a view to learning about modular forms. During the course of this chapter the book begins to ...
3
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1answer
50 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
2
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3answers
55 views

Diophantine Equatiοn $x^3=2^y+15$

I would like some help with the diophantine equation $x^3=2^y+15$ I have tried working with last digits and modular arithmetic but that hasn't got me anywhere.
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2answers
31 views

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$.

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$. I have solved the first part but second part need ...
0
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2answers
50 views

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$.

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$. My try is let $a$ be a solution of $x^2 \equiv -3 \mod p$. so $a^{p-1} \equiv 1\mod p$. This ...
1
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1answer
37 views

Primitive roots and quadratic nonresidues modulo a prime of form $2^n+1$

Let $p$ be a prime number. We call a unit $a$ in $\Bbb Z/p\Bbb Z$ a primitive root, if $\text{ord}_p(a)=p-1$. Any unit in $\Bbb Z/p\Bbb Z$ can be written as some power as some power of $a$. if $p$ is ...
1
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1answer
18 views

Evaluate the Legendre symbols (503/773) and (501/773)

Evaluate the Legendre symbols (503/773) and (501/773) my solution (501/773 ) = (((167*3))/773 ) = (167/773) * (3/773) = (773/167)*(773/3) = (105/167) * (2/3) = (3/167) * (5/167) * (7/167) * ...
2
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0answers
35 views

Bernoulli Conjecture on $B_{2^n}$

So in a recent question I was trying to prove that $2^n-1$ will never be a Carmichael number (Can a Mersenne number ever be a Carmichael number?), I was going to prove it true as long as a certain ...
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0answers
186 views
+150

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
1
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1answer
27 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
4
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0answers
73 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
2
votes
0answers
29 views

Fermat pseudo primes

Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime? In other words, can a Mersenne number be a Fermat ...
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0answers
32 views

Which basis orders [for the natural numbers] have been proven?

The set $A$ of nonnegative integers is called an additive basis of order $h$ if every nonnegative integer can be written as the sum of $h$ not necessarily distinct elements of $A$. For example, the ...