Month: July 2013

For a theoretical discussion of highly abundant numbers, I recommend a paper by Amir Akbary and Zachary Friggstad:  “”Superabundant Numbers and the Riemann Hypothesis”.  On the computational side, Keith Briggs has done extensive computations on colossally abundant numbers and their superset, the superabundant numbers. Lately, my PARI/gp computations had numbers divisible by all of the… Continue reading Conclusion of computer experiment, highly abundant numbers.

? wwwtime = 0 ms.%952 = [1787115, 942, 84, 26, 13, 9, 6, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,… Continue reading new command line

### After my last post, i worked to save arithmetic computations.### The possibility (primorial(m) replaced by primorial(m+1))### is tested tentatively for about 28 or 28 candidate### “primorial factors” which multiply to give ‘n’.### The candidate with the highest figure of merit### “wins the day” cf. previous post. ### Now, I recall the last command-line:### niter… Continue reading highly abundant numbers: new updated PARI/gp command line

? deltatime = 0 ms.%901 = [100, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,… Continue reading news on the gp command-line related to highly abundant numbers

After listening to some of the testimony of Gen. Keith Alexander (DIRNSA) and others before the US House Permanent Committee on Intelligence of June 18 2013, it occurred to me that one concern with vast metadata databases on phone calls coul arise from malevolent insiders bent on selling metadata to the highest bidder, like an… Continue reading Insider threat and Hypothetical “Aldrich Ames Version 2” scenario and NSA’s metadata (thought experiment)

This is an update, showing my latest PARI/gp “optimized” command-line to search for integers with “extreme” abundancy; one in a series of recent posts on Gronwall’s Theorem, the Theorem of Guy Robin, the Lagarias criterion for RH and the works of Ramanujan and Erdos&Alaoglu. New sample PARI/gp command line using 2 “iterations” of “outer loop”… Continue reading Superabundant numbers and number-theoretical computations with PARI/gp

There was a hearing before the US House Permanent Select Committee on Intelligence on the topic of NSA Data Collection Programs.  This occurred on June 18 2013 and is on C-SPAN as 3 hours of audio/video. So, I think it might be worth watching. http://www.c-spanvideo.org/program/AgencyOp

The memorandum in support of a motion on shielding FISA/FISC proceedings from the defendant says in part that,    the court “may disclose to the aggrieved person, under appropriate security procedures and protective orders, portions of the application, order or other materials relating to the surveilance only where such disclosure is necessary to make an… Continue reading memorandum in US v. Basaaly Saeed Moalin

This is what I should have written: I’d say the Lagarias criterion is in the same vein as Gronwall’s Theorem as well as Robin’s inequality and Robin’s Theorem (after Guy Robin, 1984). Reference:  Wikipedia article on Jeffrey Lagarias, at http://en.wikipedia.org/wiki/Jeffrey_Lagarias#Career   Using arrays (vectors) in PARI/gp containing the prime exponents of a number for the… Continue reading Correction to last post

Abundant here refers to the the sum of divisors function sigma(n) relative to the size of n, a large positive integer. J. Lagarias found a simple criterion equivalent to the Riemann Hypothesis involving “extremely large” values of sigma(n) relative to n.  I’d say the criterion is in the same vein as Gronwall’s Theorem as well… Continue reading PARI/gp code to find abundant products of primorials