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Author Archives: apgoucher
A combinatorial proof of Houston’s identity
Robin Houston recently discovered a rather interesting formula for the determinant of an nbyn matrix. In particular, the formula improves upon the best known upper bound for the tensor rank of the determinant (viewed as a multilinear map which takes … Continue reading
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Tetrational machines
A pair of people called Pavel have independently developed remarkable automata that last recordbreakingly long before halting. In both cases, the number of timesteps that it takes for each automaton to halt is so large that it cannot be written … Continue reading
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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
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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
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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
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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 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
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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 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
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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
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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
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