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Results tagged with nt.number-theory
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user 2290
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Accepted
What is the order of a in (Z/nZ)*?
You seem to have been given some misinformation so I'll answer this question although I think it is elementary. You want to find the order of $a$ modulo $n$. The prime factorization of $a$ is largely …
- 30.1k
3
votes
known methods for solving diophantine systems?nt
You must not have looked very hard. This will get you started:
Nigel P. Smart. The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts 41. Cambridge University …
- 30.1k
1
vote
Accepted
multiplicative order of 2 mod p^N
I don't think this is known. You might want to have a look at:
A. Granville, Refining the conditions on the Fermat quotient,
Mathematical Proceedings of the Cambridge Philosophical Society, 98 (1985) …
- 30.1k
11
votes
Where can I find information about Lagrange's Theorem with certain squares left out?
There is a formula (due to Jacobi) for the number of representations of an integer as a sum of four squares and estimates for the number of representation of an integer as a sum of three squares (e.g. …
- 30.1k
7
votes
Accepted
Four polynomials representing all integers modulo m
For a prime $p>2$, fix a nonsquare $c$. If you find $y$ such that $y/3$ is a non-square (i.e. $y/3=cx^2, x\ne0$) and $y/3 - 1/9 = cz^2, z\ne 0$, then $y$ is not represented by the first two polynomial …
- 30.1k
5
votes
Expansions by cube roots of 1 (mod n)
If $n$ is prime and $\equiv 1 \mod 3$, you don't need $r^2$ since $r^2+r+1=0$. So you can take $a=0$ and you can have $b,c = O(n^{1/2})$, since if two such numbers are congruent mod $p$, then their di …
- 30.1k
4
votes
Accepted
Counting the number of prime triplet
Under your assumptions $p,q,r$ are all about size $x= 10^{l/3}$. The congruence conditions are basically independent so you'd get about $(x/\log x)^3(\phi(m)-1)/\phi(m)^3$. There may be a constant in …
- 30.1k
18
votes
Accepted
Inverse problem for zeta functions of curves over finite fields
Tate and Honda show that almost all polynomials like that are the L-function of an abelian variety over the finite field. The problem with curves is much harder and it's open (for genus g>2). One nece …
- 30.1k
2
votes
Accepted
Upper bounds on the number of representation of a natural number as a sum of $s$ positive $k...
There are $O(x^{1/k})$ $k$-th powers of size at most $x$, so there are $O(x^{s/k})$ sums of $s$ $k$-th powers of size at most $x$ but only $x$ integers of size at most $x$ so some integer at most $x$ …
- 30.1k
5
votes
Is there a finite set of primes such that if K over Q is completely split at all those prime...
No. If d is a square modulo p for all p in S, then all p in S split in $\mathbb{Q}(\sqrt d)$.
- 30.1k
2
votes
Restricting the Lindelöf hypothesis to critical line integer values
I doubt it. The zeta function has lots of zeros on the critical line and, for those zeros of the form $1/2+ i\gamma, \zeta(1/2+i\gamma)=0 = O(|\gamma|^{\epsilon})$, trivially. So, stepping on an infin …
- 30.1k
1
vote
Regular maps between quasi-projective varieties defined over a global field
Let
$V = \{x \in \mathbb{P}^n |\forall i \in I, f_i(x) = 0, \exists j \in J,g_j(x) \ne 0\}$
and $W = \{y \in \mathbb{P}^m |\forall i \in I', f'_i(x) = 0, \exists j \in J', g'_j(x) \ne 0 \} $
be quasi …
- 30.1k
5
votes
When is the sum of two quadratic residues modulo a prime again a quadratic residue?
It is easy to write this number (of $a$ such that $a,a+1$ are squares) in terms of the number of solutions of $x^2-y^2=1$. This is a conic which has $p+1$ projective points over the field of $p$ eleme …
- 30.1k
-1
votes
Accepted
Mersenne numbers represented by a Quadratic polynomial in two variables
I think this was proved by Iwaniec (for any quadratic in two variables satisfying the obvious necessary conditions).
- 30.1k
6
votes
Using Deligne's theorem to estimate exponential sums
Theorem 8.4 of Deligne's paper (Weil I) gives what you want with $C_f=(d-1)^2, d = \deg f$, provided the homogeneous part of degree $d$ of $f$ defines a "smooth variety" in $\mathbb{P}^1$, which (in o …
- 30.1k