Author Archives: apgoucher

29-year-old 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

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Training a random Gaussian generator

I’ve spent the last couple of months tackling the problem of designing an algorithm to rapidly generate high-quality normally-distributed pseudorandom numbers on a GPU. Whilst this may seem quite pedestrian, it turned out to be much more interesting than I’d … Continue reading

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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

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Hamming backups: a 2-of-3 variant of SeedXOR

EDIT (2021-10-14): 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

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An efficient prime for number-theoretic 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

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Hamming cube of primes

Given two nonnegative integers, m and n, we say that they are Hamming-adjacent if and only if their binary expansions differ in exactly one digit. For example, the numbers 42 and 58 are Hamming-adjacent because their binary expansions 101010 and … Continue reading

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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 order-preserving* bijection between the nonnegative reals [0, ∞) and ‘Baire space‘, , the space … Continue reading

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One-way permutations

One-way 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

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Cyclotomic fields

The nth cyclotomic field is the field generated by a primitive nth root of unity, ζ. It is an example of a number field, consisting of algebraic numbers, and its dimension is φ(n) when regarded as a vector space over … Continue reading

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Urbit for mathematicians

The three months since the last cp4space article have largely been spent learning about an interesting project called Urbit. My attention was drawn to its existence during a talk by Riva-Melissa Tez (formerly of Intel) on the importance of continued … Continue reading

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