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Results tagged with prime-numbers
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user 2024
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
10
votes
Stronger versions of Wilson's Theorem
Short expansion of my comment on what's immediately achievable when assuming the $abc$-conjecture.
Proporistion. Assuming the $abc$-conjecture, we have $$v_p((p-1)!+1)=o(p).$$
Proof. One formula …
- 4,593
1
vote
0
answers
195
views
Lower bound on number of smooth values of polynomial at primes
Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth \}|} …
- 4,593
17
votes
5
answers
3k
views
Families of number fields of prime discriminant
When I am testing conjectures I have about number fields, I usually want to control the ramification, especially minimize to a single prime with tame ramification. Hence, I usually look for fields of …
- 4,593
2
votes
Accepted
Calculating the infinite product from the Hardy-Littlewood Conjecture F
This problem was studied by a few, and the ideas involve too much latex to write here. Mainly there are ideas of transforming to crazy weighted sums and then use ERH to bound errors from crazier integ …
- 4,593