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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
Apr 18 |
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When has the Borel-Cantelli heuristic been wrong?
Another example of (probable) failure in the presence algebraic structure: Pomerance has a heuristic, described at oddperfect.org/pomerance.html, predicting no odd perfect numbers. The same heuristic could be adapted to suggest there are only finitely many even perfect numbers --- but of course we "know better" here, by Euclid. |
Mar 26 |
awarded | Nice Answer |
Mar 26 |
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Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power
Let me supplement my previous comment by saying that it's not realistic at this time to expect definitive results for fixed $a$. For example, consider the equation $\sigma(n) = 6^b$. Probably there are infinitely many solutions $n,b$, since if $p=2\cdot 6^m-1$ is prime, then $n=2p$ gives rise to a solution. On the other hand, it seems hopeless to prove at this time that there are infinitely many solutions. For then there are infinitely many $p$ with $p+1$ a $3$-smooth number. We don't know anything nearly this strong! The ``smoothest'' we can get $p+1$ is $p^{0.2931}$ (Baker-Harman)! |
Mar 25 |
answered | Does there exist an integer that is both solitary and almost perfect? |
Mar 25 |
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Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power
Here is a comment on the fixed $a$ problem: One can show that there are only finitely many proper prime powers $p^e$ for which $\sigma(p^e)$ is supported on a given finite set of primes. See, e.g., the book of Shorey and Tijdeman on Exponential Diophantine Equations. This means that the problem in some sense reduces to understanding the shifted primes $\sigma(p)=p+1$. For instance, $\sigma(n) = 2^b$ has infinitely many solutions $(n,b)$ if and only if there are infinitely many Mersenne primes. |
Mar 24 |
answered | Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power |
Feb 11 |
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Waring problem for binomial coefficients (generalization of Gauss' Eureka Theorem)
As Gjergji has pointed out, this is a consequence of a general theorem of Kamke. A modern reference is Nathanson's Elementary methods in Number Theory; see Chapter 11, especially Theorems 11.10 and 11.12. |
Feb 1 |
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Surveys of the items of Erdős' “toolbox”
The recent volume that came out of the Erdős centennial conference might also be of interest to you: springer.com/new+%26+forthcoming+titles+%28default%29/book/… |
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? |