
Recent Posts
Subscribe to Complex Projective 4Space
Join 2,954 other subscribersArchives
 July 2023
 March 2023
 February 2023
 January 2023
 October 2022
 September 2022
 July 2022
 June 2022
 May 2022
 January 2022
 December 2021
 September 2021
 July 2021
 June 2021
 May 2021
 February 2021
 January 2021
 December 2020
 November 2020
 October 2020
 September 2020
 August 2020
 July 2020
 June 2020
 May 2020
 June 2019
 May 2019
 March 2019
 November 2018
 September 2018
 July 2018
 June 2018
 May 2018
 April 2018
 March 2018
 February 2018
 November 2017
 October 2017
 November 2016
 May 2016
 March 2016
 February 2016
 December 2015
 September 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
 May 2014
 April 2014
 March 2014
 February 2014
 January 2014
 December 2013
 November 2013
 October 2013
 September 2013
 August 2013
 July 2013
 June 2013
 May 2013
 April 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
Recent Comments
 pzqalex on Iterated Aut revisited
 NoLongerBreathedIn on The Osmiumlocks Prime
 Konkhra on Miscellaneous discoveries
 Gerry Myerson on Rational approximations to Platonic solids
 Lucas Brown on Threelds
Author Archives: apgoucher
Infinitely many rational dodecahedra
Thomas Blok and David Madore have recently made significant progress on the problem of finding rational dodecahedra inscribed in the unit sphere, culminating in an infinite parametric family of solutions. In particular, Thomas began with the constrained version of the … Continue reading
Posted in Uncategorized
6 Comments
Threelds
A field F consists of two compatible Abelian groups — an additive group on F and a multiplicative group on F \ {0} — such that multiplication distributes over addition. In certain cases, though, this multiplicative group can be the … Continue reading
Posted in Uncategorized
5 Comments
29yearold Conway conjecture settled
Ilkka Törmä and Ville Salo, a pair of researchers at the University of Turku in Finland, have found a finite configuration in Conway’s Game of Life such that, if it occurs within a universe at time T, it must have … Continue reading
Posted in Uncategorized
22 Comments
Training a random Gaussian generator
I’ve spent the last couple of months tackling the problem of designing an algorithm to rapidly generate highquality normallydistributed pseudorandom numbers on a GPU. Whilst this may seem quite pedestrian, it turned out to be much more interesting than I’d … Continue reading
Posted in Uncategorized
Leave a comment
Involutions on a finite set
Suppose that R is the (assumed to be finite) set of solutions to a certain problem, and you’re interested in determining the parity of R. The following proof strategy works surprisingly often, namely in at least two different scenarios, and … Continue reading
Posted in Uncategorized
7 Comments
Hamming backups: a 2of3 variant of SeedXOR
EDIT (20211014): I’ve written a reference implementation of the Hamming backup idea introduced in this article. SeedXOR is an approach for splitting a Bitcoin wallet seed, adhering to the BIP39 standard, into N ‘parts’ (each the same size as the … Continue reading
Posted in Uncategorized
1 Comment
An efficient prime for numbertheoretic transforms
My new favourite prime is 18446744069414584321. It is given by , where . This means that, in the finite field , 2^32 functions as a primitive 6th root of unity, and therefore 2 is a primitive 192nd root of unity. … Continue reading
Posted in Uncategorized
8 Comments
Hamming cube of primes
Given two nonnegative integers, m and n, we say that they are Hammingadjacent if and only if their binary expansions differ in exactly one digit. For example, the numbers 42 and 58 are Hammingadjacent because their binary expansions 101010 and … Continue reading
Posted in Uncategorized
2 Comments
Determinacy
I’d like to take this opportunity to highly recommend Oscar Cunningham’s blog. One of the posts, entitled A Better Representation of Real Numbers, describes an elegant orderpreserving* bijection between the nonnegative reals [0, ∞) and ‘Baire space‘, , the space … Continue reading
Posted in Uncategorized
Leave a comment
Oneway permutations
Oneway permutations are fascinating functions that possess a paradoxical pair of properties. They are efficiently computable functions φ from [n] := {0, 1, …, n−1} to itself that are: Mathematically invertible, meaning that φ is a bijection; Cryptographically uninvertible, meaning … Continue reading
Posted in Uncategorized
Leave a comment