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Results tagged with nt.number-theory
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user 7076
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
Asymptotic of $Lcm(\binom{2k}k)_{1\le k\le n}$
Kummer's theorem implies the formula:
$$\mathrm{LCM}(\binom{2}{1},\binom{4}{2},\dots,\binom{2n}{n}) = 2^{\lfloor \log_2(n+1)\rfloor} \cdot \prod_{3\leq p\leq 2n} p^{\lfloor \log_p(2n-1)\rfloor},$$
whe …
- 29.5k
2
votes
Accepted
Characters of integers of $4p^2q^2-(p\pm q)^2$ form
It can be seen that both forms can be factored as differences of squares.
Correspondingly, $n$ can be expressed in these forms iff there exists a divisor $d\mid n$ such that $d$ and $\frac{n}{d}$ have …
- 29.5k
2
votes
Squares of the form $2^j\cdot 3^k+1$
It is also possible to reduce to 9 Mordell equations. Denoting $x := 2^{[i/3]}3^{[k/3]}$, we get 9 equations:
$$d x^3 + 1 = y^2$$
indexed by $d\mid 2^23^2$.
Equivalently,
$$(dy)^2 = (dx)^3 + d^2,$$
wh …
- 29.5k
8
votes
Partitioning $2p$, subject to a divisibility condition.
It is possible. Take $p=61$ and $t_1=2$, $t_{120}=1$. Then $l=113$ and $1^{t_1}\cdot t_1!\cdot 120^{t_{120}}\cdot t_{120}!=240$ divides $(2p-l)!=9!=362880$.
Another example: $p=673$, $t_1=t_{672}=2$. …
- 29.5k
4
votes
s(n) = kn or s(n) = n/k?
This is related to some open questions in number theory. See http://oeis.org/A134639 and references/links there.
Conjecture 2 is incorrect, the smallest counterexample is given by $$n=154345556085770 …
- 29.5k
3
votes
Need to require a assertion for two subsets of natural numbers
Consider the set
$$T = \{ (x,y)\ :\ 1\leq x\leq Q,\ 1\leq y\leq P,\ Px+Qy\leq PQ\}.$$
It is easy to see that
$$S = \sum_{(x,y)\in T} (Px+Qy).$$
Switching to double summation in $S$, we get
$$S = \sum …
- 29.5k
4
votes
Is there an asymptotic formula that describes the correlation of multiplicative inverses in ...
Finding asymptotic for $S(X)$ amounts to estimation of $\sum_{(a,q)=1} aa^*$ for any given $q$.
From the rearrangement inequality, it follows that
$$\frac{1}{2}q^2\varphi(q) - T = \sum_{(a,q)=1} a(q …
- 29.5k
1
vote
Euler's theorem for Tetration, Pentation, etc. (Superexponentiation)
Suppose that for a positive integer $m$, there exists a positive integer $a>1$ such that $(a,m)=1$ and $\mathrm{rad}(\mathrm{ord}_m(a))\not|a$. Then $f_2(m)>0$ does not exists.
(Here $\mathrm{ord}_m( …
- 29.5k
4
votes
Accepted
Restriction of Jacobi's four-squares theorem
By inclusion-exclusion principle, the number of representations of $n$ as the sum of squares of four nonzero integers equals:
$$\sum_{k=0}^4 \binom4k (-1)^k r_{4-k}(n).$$
Formulae for $r_k(n)$ are giv …
- 29.5k
4
votes
how can I minimise (n * y) (mod x) for known x and y, and for a given range of n?
What I'm going to say is somewhat similar to Roland's answer but more precise in the case when the range for $n$ is given in the form of upper bound, i.e., $0 < n < N$.
Notice that $ny\bmod x = ny - …
- 29.5k
3
votes
Solutions of the equation $2^{q-1} \equiv q \pmod {4q^2+1 }$ where $q$ is an odd prime
I've checked all primes $q$ below $10^{11}$ with no new solutions found (even if the primality of $p=4q^2+1$ is relaxed).
Notice that $2^{q-1}\equiv q\pmod{4q^2+1}$ implies $2^{2q}\equiv 4q^2\equiv - …
- 29.5k
2
votes
A sum involving mod(n) arithmetic
I believe the formula $|A|=\frac{\varphi(q)}{\mathrm{lcm}(\varphi(d_1),\varphi(d_2))}$ is not quite correct. In particular, for $q=15$, $d_1 = 3$, $d_2=5$, $a=1$, this formula gives $|A|=2$, while, in …
- 29.5k
8
votes
Elementary divisibility problem
Zsigmondy theorem states that for any $p>1$, $a^p+b^p$ has a primitive prime factor, except when $\{a,b\}=\{1,2\}$. Such prime does not divide $a+b$. Hence, $a^p+b^p$ cannot divide $(a+b)^p$ unless it …
- 29.5k
1
vote
The existence of solutions of a system of indeterminate equations
I expect that in many cases the following construction will do the job. By Fermat polygonal number theorem, we have representation of $\frac{m(m-3)}4$ as the sum of 3 triangular numbers:
$$ \frac{m(m- …
- 29.5k
2
votes
Accepted
The existence of solutions of a system of indeterminate equations
In the comments OP proposed a greedy algorithm to represent a given positive integer $A$ as the sum of triangular numbers whose indices sum to $m$, and applied it to $A = \frac{m(m+1)}4$. I will prove …
- 29.5k