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These are all questions I would never even have asked myself until that incident with Don. Every day, my friend Don and I would see who could trip each other the most times. But then one day I tripped him and he fell and broke his jaw. He looked up and, with slurred speech, said, “I guess you win.” But what did I win? I didn't win anything, and you know why? Because I forgot to make a bet with him. But something else was wrong, and I knew it. Why did I want to trip Don in the first place? To show how clever I was, or how brave, or how successful? Yes, all those things. So I guess that answers that. -- Jack Handey
Dec 24 |
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Ruth-Aaron triples, etc
It seems that, contra Hoffman, Erdos never claimed to prove the infinitude of Ruth--Aaron pairs. After the original Ruth--Aaron paper appeared in the Journal of Recreational Mathematics, Pomerance learned of Erdos's interest from a letter (not a phone call). In the letter, Erdos says he can prove that the Ruth--Aaron numbers have density $0$. Far from claiming a proof that there are infinitely many Ruth--Aaron numbers, Erdos says that problem "seems hopeless" ! The original letter is available online --- see the very first scan at cah.utexas.edu/collections/math_erdos.php |
Nov 24 |
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Siegel Walfisz Theorem for algebraic number fields
An analogue of of Siegel--Walfisz for Hecke characters can be found in the following paper: Goldstein, Larry Joel: A generalization of the Siegel-Walfisz theorem. Trans. Amer. Math. Soc. 149 1970 417–429. |
Nov 10 |
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A conjecture of Erdos on consecutive differences of primes
The problem is solved by recent work of Banks--Freiberg--Turnage-Butterbaugh. Using the method of Maynard--Tao, they show that the sequence of e_k's contains arbitrarily long runs of 0's, as well as arbitrarily long runs of 1's. Hence, the sum has a nonperiodic binary expansion and so represents an irrational number. |
Oct 30 |
awarded | Nice Answer |
Oct 25 |
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Effective version of the Bombieri-Vinogradov theorem
For effective variants of Bombieri--Vinogradov, see e.g. this paper of Akbary and Hambrook: cs.uleth.ca/~akbary/Akbary-Hambrook.pdf or Lemma 5.2 of this paper of Lenstra and Pomerance: math.dartmouth.edu/~carlp/PDF/complexity12.pdf |
Oct 25 |
awarded | Nice Answer |
Oct 24 |
answered | Is every number the sum of two cubes modulo p where p is a prime not equal to 7? |
Oct 13 |
revised |
Prime factors of the members of a certain recurrence
deleted 12 characters in body |
Oct 13 |
answered | Prime factors of the members of a certain recurrence |
Sep 24 |
awarded | Autobiographer |
Sep 23 |
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Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
Actually [1] predates the Alladi--Erdos paper; see: A. E. Brouwer, Two number theoretic sums, Mathematisch Centrum, Amsterdam, 1974, Mathematisch Centrum, Afdeling Zuivere Wiskunde, ZW 1974. |
Aug 24 |
answered | A family Mersenne composite numbers? |
Aug 12 |
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Divisor sums over values of binary forms of primes
@Gerry: Computing asymptotics for the partial sums of $\tau(p+a)$, with $a$ fixed, is known as the Titchmarsh divisor problem. This was solved by Linnik, but nowadays can be done by the Brun--Titchmarsh and Bombieri--Vinogradov theorems. As for $\tau(x^2+y^2)$, it's maybe more natural to consider the sum extended over all pairs $x,y$ with $x^2+y^2 \le n$. That can be attacked by applying a mean-vale theorem of Wirsing to the function $n\mapsto \tau(n) r(n)$, where $r(n) = \frac{1}{4} \#\{(x,y):x^2+y^2=n\}$. (Note that $r$ is multiplicative.) |
Aug 10 |
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How many integers divide a number that involves just three non-zero digits?
Not an answer to your question, but Skalba has shown that if the order of 2 mod p exceeds $p^{4/5}$, then p divides some number of the form $2^a + 2^b+1$. We don't know that Skalba's criterion holds for a full density of primes of $p$, but we expect it to do so (e.g., that follows from GRH). He also shows that if $2^m-1$ has fewer than $\log{m}/\log{3}$ prime divisors (counted with multiplicity), then there is a prime dividing $2^m-1$ not dividing any number of the form $2^a+2^b+1$. See: Skałba, Mariusz, Two conjectures on primes dividing $2^a+2^b+1$, Elem. Math. 59 (2004), no. 4, 171–173. |
Jul 24 |
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Examples of famous 'workhorse' theorems
Surely "Brun's sieve" qualifies. |
Jul 22 |
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Constructing quintic number fields with certain splitting behaviour
Doesn't the hypothesis in Theorem 1.3 only require what you said for all large enough p? So why can't you (or can you?) run your argument up to an arbitrarily large finite height to get an arbitrarily small upper density, and hence a density of 0? |
Jul 21 |
revised |
Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions
corrected spelling of "specimen" |
Jul 21 |
answered | Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions |
Jul 18 |
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Proof of equidistribution theorem for exponential coefficients
Asking that $\{2^n \alpha\}_{n \ge 1}$ be equidistributed mod $1$ is exactly equivalent to asking that $\alpha$ be normal in base $2$. Not a single irrational algebraic number is known to be $2$-normal; for some partial results, see this paper of Bailey, Borwein, Crandall, and Pomerance: crd.lbl.gov/~dhbailey/dhbpapers/algebraic.pdf |
Jul 17 |
awarded | Enlightened |