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Results tagged with nt.number-theory
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user 19844
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
1
answer
333
views
Alternate Definition(s) for (Ir)regular Primes
Are there any definitions for (ir)regular primes which do not use class number divisibility or Bernoulli numbers? For reference, Wikipedia gives both the first definition (their primary one) and the s …
- 1,055
3
votes
3
answers
11k
views
prime factors of x^2 - 2
In one of his letters to Frenicle, Fermat stated the proposition that no prime of the form $q^2+2$ can divide any number of the form $x^2-2$.
Is there a known proof of this statement?
If not, how wou …
- 1,055
3
votes
Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?
I realise this is a massive revival/refresh of this question, but I just found a completely elementary solution of this problem, outlined in this MSE question and my own answer.
Does anyone know if t …
- 1,055
3
votes
On integers as sums of three integer cubes revisited
Perhaps if you start with my three-rational-cubes identity
$$
ab^2 = \biggl(\frac{(a^2+3b^2)^3+(a^2-3b^2)(6ab)^2}{6a(a^2+3b^2)^2}\biggr)^{\!3}
- \biggl(\frac{(a^2+3b^2)^2-(6ab)^2}{6a(a^2+3b^2)}\ …
- 1,055
6
votes
4
answers
571
views
seeking an integer parameterization for A^2+B^2=C^2+D^2+1
I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation
$A^2+B^2=C^2+D^2+1$,
analogous to the classical parameterization of the Pythagorean equatio …
- 1,055
0
votes
1
answer
411
views
Given a generating function with "zeros", can one derive the function for ONLY the "zeros"?
If I have an generating function (GF) --- ordinary or exponential --- defining a series with at least one coefficient equal to zero, is there a general method to find the "inverse GF", i.e., the GF de …
- 1,055
2
votes
1
answer
265
views
Does the congruence $a^p \equiv 1 \pmod{b^p}$ with prime $p \ge 5$ force $b \le p$?
I'm considering the congruence in the title, i.e.,
$$a^p \equiv 1 \pmod{b^p},$$
where $a \ge b \ge 1$ are positive integers and $p$ is an odd prime.
For $p=3$, a brute-force computer search found man …
- 1,055
17
votes
3
answers
2k
views
Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b...
Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < 2\beta$. …
- 1,055
1
vote
0
answers
164
views
For which recurrence relations is it decidable whether a formal power series has a maximal z...
In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”.
…
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1
vote
Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$
Inspired by @PeterMueller, I believe I found a proof that $r = 3$.
Because of how this equation was obtained in the first place, I can assume $s \ge 2$ is even, and $r \ge s+1$ is odd. Writing $s=2v$ …
3
votes
1
answer
401
views
Proving conditions on $(r+s)^2 \mid (4r^4+1)$, related to Pell oblongs
While working on another problem (Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$), I came across a question which seems to be of [semi-] independent interest.
Conjecture. If $ …
- 1,055
3
votes
1
answer
369
views
Is there an easy proof of this equation related to simultaneous Pell equations?
Working with the famous Baker-Davenport system of simultaneous Pell equations
\begin{align}
3x^2-2 &= y^2, &
8x^2-7 &= z^2, \qquad(\star)
\end{align}
I am left, after a series of substitutions and …
- 1,055
17
votes
2
answers
1k
views
Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?
While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction
$$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$
Based on what …
- 1,055
13
votes
3
answers
3k
views
Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$
I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are
\begin{equation}
(r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, -1) …
13
votes
Fermat's proof for $x^3-y^2=2$
A completely elementary proof can be found on page 561 of the Nov 2012 edition of The Mathematical Gazette, where a descent mechanism first used by Stan Dolan in the March 2012 edition is adapted (as …
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