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Summary

Ordinary probabilities are real numbers, and ordinary entropy is a real number too. Building on ideas of Kontsevich, we develop an analogue of entropy in which both the probabilities and entropy itself are integers modulo a prime number p. While the formula for entropy mod p is quite unlike the formula for real entropy, we prove characterization theorems for entropy mod p and information loss mod p that are very closely analogous to the theorems over the real numbers, thus justifying the definition. We also establish a sense in which entropy mod p is the residue mod p of real entropy.

In 1950 Erdös proved that if $x\equiv2\,036\,812\ (\mo5\,592\,405)$ and $x\equiv3\ (\mo62)$, then $x$ is not of the form $2^n+p$, where $n$ is a non-negative integer and $p$ is a prime. In this note we present a theorem on integers of the form $2^n+cp$, in particular we completely determine all those integers $c$ relatively prime to $5\,592\,405$ such that the residue class $2\,036\,812(\mo5\,592\,405)$ contains integers of the form $2^n+cp$.

AMS 2000 Mathematics subject classification: Primary 11P32. Secondary 11A07; 11B25; 11B75

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11 - Entropy Modulo a Prime

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A NOTE ON INTEGERS OF THE FORM 2n+cp

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11 - Entropy Modulo a Prime

Summary

A NOTE ON INTEGERS OF THE FORM 2n+cp