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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
1d

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RuthAaron triples, etc
It seems that, contra Hoffman, Erdos never claimed to prove the infinitude of RuthAaron pairs. After the original RuthAaron 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 RuthAaron numbers have density $0$. Far from claiming a proof that there are infinitely many RuthAaron 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 SiegelWalfisz for Hecke characters can be found in the following paper: Goldstein, Larry Joel: A generalization of the SiegelWalfisz 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 BanksFreibergTurnageButterbaugh. Using the method of MaynardTao, 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 BombieriVinogradov theorem
For effective variants of BombieriVinogradov, see e.g. this paper of Akbary and Hambrook: cs.uleth.ca/~akbary/AkbaryHambrook.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 AlladiErdos 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 BrunTitchmarsh and BombieriVinogradov 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 meanvale 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 nonzero 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 $2^m1$ has fewer than $\log{m}/\log{3}$ prime divisors (counted with multiplicity), then there is a prime dividing $2^m1$ 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 