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 Nov 18 comment Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers Have you seen this recent arXiv preprint? arxiv.org/abs/1509.02590 Nov 8 comment What is the asymptotic growth rate of the product of divisor function up to n Well, $f(n)=\log\tau(n)$ is an additive function; that is, $f(n) = \sum_{p^k\parallel n} f(p^k)$. Now insert this expression for $f$ into $\sum_{n \le x}f(n)$ and reverse the order of summation. I believe in this example, one finds that the main term in the asymptotic is $x \sum_{p \le x} f(p)/p$, and this is $\sim (\log 2) x\log\log{x}$, as $x\to \infty$ (Alternatively, note that $\tau(n)$ is between $2^{\omega(n)}$ and $2^{\Omega(n)}$, and use the known results --- as found in Hardy and Wright, for example --- on the partial sums of $\omega$ and $\Omega$.) Oct 31 answered What is known about the largest prime divisor of the product of $k$ consecutive integers? Oct 25 comment multiplicative functions of powers There's a beautiful expository paper by Moree and Cazaran that touches on this; see cl.ly/0V012j0O1a1B Oct 25 answered multiplicative functions of powers Oct 25 comment multiplicative functions of powers Perhaps I don't understand the question. But $\tau(n^k)$ is a nonnegative multiplicative function. And there are very general theorems giving asymptotics for partial sums of nonnegative multiplicative functions; e.g., there is a beautiful theorem of Wirsing that immediately applies to this question. See Wirsing, Eduard Das asymptotische Verhalten von Summen über multiplikative Funktionen. (German) Math. Ann. 143 1961 75–102. (But I think Wirsing is "overkill" here, in the sense that Brad Rodgers's suggestion above also works and is simpler.) Oct 19 answered How often is $2^n-1$ a number with few divisors? Oct 5 answered Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$? Sep 28 comment Does the equation $241+2^{2s+1}=m^2$ have a solution? If anyone is wondering, the argument Mike Bennett refers to is worked out in math.dartmouth.edu/~carlp/2tokmmminus1v8.pdf . See Theorem 1. Aug 15 answered Prime divisors of values of a polynomial on an infinite set Jul 31 answered Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$ Jul 28 comment Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial An alternative is to use some of the theory of Beurling primes. See alpha.math.uga.edu/~pollack/beurling.pdf Jul 28 comment Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial A quick comment: One can derive the estimate $\sum_{p \le x} \nu_F(p)/p = \log\log{x} + O(1)$ without worrying about zeros of $\zeta_K(s)$ on $\Re(s)=1$; e.g., apply the Tauberian theorem mentioned as Proposition 5 in alpha.math.uga.edu/~pollack/eulerprime.pdf to $\log \zeta_K(s)$. Everything needed to check the hypotheses is in Hecke's algebraic number theory book (and goes back to Dedekind, I think). Jul 17 awarded Yearling Jun 25 comment Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L I don't have an account on Math Stackexchange at the moment, but the conjecture of Wythagoras that motivated your question follows quickly from Theorem 1 in this paper of Erdos--Pomerance--Schmutz: math.drexel.edu/~eschmutz/PAPERS/lambda.pdf Jun 6 comment When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$? Have you seen Proposition 3.8 and Example 3.16 in esc.fnwi.uva.nl/thesis/centraal/files/f310232185.pdf ? May 25 comment On the number of consecutive divisors of an integer Here's a PDF: renyi.hu/~p_erdos/1978-26.pdf May 25 answered On the number of consecutive divisors of an integer May 14 answered On the natural density of almost perfect numbers May 10 revised Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite? deleted extraneous plus sign