All Questions
17 questions
1
vote
1
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240
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The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
2
votes
0
answers
146
views
Reference for accelerated sum to compute the Meissel-Mertens constant
The Meissel-Mertens constant
$$ B_1 = \lim_{n \to \infty} \left(\sum_{p \leq n} \frac{1}{p} - \log\log n\right) $$
has the series representation
$$
\begin{equation} \tag{1}
B_1 = \gamma + \sum_{n=2}^{...
- 451
6
votes
0
answers
200
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Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$
It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$
for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
- 20.2k
3
votes
0
answers
97
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Study of relative class number of 'non-abelian' CM field by using L-functions
I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields.
So I'm looking for some references to learn the techniques that can be useful.
So far, I ...
- 1,013
3
votes
0
answers
73
views
Reference: Asymptotic bit-complexity of algebraic operations and transcendental functions
This question is a reference request. Does anyone know of a reference that lists the asymptotic bit-complexity of algebraic operations and transcendental functions implemented on a Turing machine that ...
- 131
1
vote
0
answers
77
views
Digit summation of squared numbers
In olympiad teaching period, we have a session that students must try to design a good problem for others. Many times we arrive to good questions but sometimes there are some challenges. In one of our ...
- 4,784
3
votes
1
answer
137
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Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
- 13.9k
3
votes
1
answer
510
views
Yet another question on sums of the reciprocals of the primes
I recall reading once that the sum $$\sum_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$
is less than $4$.
Does anybody here know what the ultimate source of this claim is?
Please, let me ...
- 8,842
3
votes
1
answer
108
views
Is coprimality in $NC$?
Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
- 13.9k
4
votes
1
answer
288
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Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
- 13.9k
0
votes
0
answers
257
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Hercules and the Hydra with time constraints
The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
- 2,302
2
votes
2
answers
257
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Reference request for function by which to compute coefficients of continued fraction of algebaic number
The simple continued fraction is in the form
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
- 3,094
9
votes
2
answers
647
views
On bounds for idoneal integer
What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
user76479
1
vote
1
answer
325
views
Error term for prime harmonic
What is known about the asymptotic behavior of
$$
f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1?
$$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega_\pm$,...
- 9,114
8
votes
3
answers
2k
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Numerical evaluation of the Petersson product of elliptic modular forms
It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
4
votes
0
answers
369
views
Reducing factoring prime products to factoring integer products (in average-case)
My question is about the equivalence of the security of various candidate one-way functions that can be constructed based on the hardness of factoring. (This question has been asked also in the CS ...
- 41
12
votes
3
answers
881
views
What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?
Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute
the parity of p(n) quickly for quite large n. In:
On the distribution of parity in the partition function, ...
- 5,422