Peter Shiu

A Footnote to the Three Gaps Theorem

Abstract Let α > 0 be an irrational number and n > 1. The three gaps theorem states that, when the fractional parts of α, 2α, …, nα are arranged in an ascending order, the gaps between successive terms may take at most three distinct values. We give a criterion for the exact number of distinct values being taken.

The gaps between sums of two squares

Problems concerning the set of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4 k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W , namely that they have the form PQ 2 , where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

Finding a missing digit

‘There is a simple argument which shows that, for any positive real number x , the digit 7 appears in the decimal representation of one of the numbers x, 2x, … , 79x . I spent a wet Saturday afternoon showing that 79 can be reduced to 42, which is best possible.’ Some thirty years ago this interesting problem was given to me verbally by David Masser, who was a reader in mathematics at Nottingham University. I solved the problem, over several days, and forgot about it. While tidying up papers in retirement I found the notes on the problem, and when I mentioned this to David he too had forgotten about it until he searched through his notes. He now encourages me to publicise the problem, which is presented here with a small extension.

A lower bound for the prime counting function

Let π ( x ) count the primes p ≤ x , where x is a large real number. Euclid proved that there are infinitely many primes, so that π ( x ) → ∞ as x → ∞; in fact his famous argument ([1: Section 2.2]) can be used to show that There was no further progress on the problem of the distribution of primes until Euler developed various tools for the purpose; in particular he proved in 1737 [1: Theorem 427] that

93.20 Egyptian fraction representations of 1 with odd denominators

The two optimisation problems associated with the representation of 1 by Egyptian fractions with odd denominators are solved. The unique solution with denominators up to 105 is given by 3, 5, 7, 9, 11, 33, 35, 45, 55, 77, 105. There are five solutions when only nine denominators are used. 2000 Mathematical Subject classification: 11D68

Plato's hidden theorem on the distribution of primes

A conjecture by the late professor Andreas Zachariou of the Department of Mathematics of the University of Athens that a passage in Book 5, 737e, 738 of Platos “Laws” is in fact a “hidden theorem” concerning the arrangement of prime numbers is revisited and a proof of this remarkable result given.

A Diophantine Property Associated with Prime Twins

We show that there is initial order rather than chaos for the solution to a Diophantine difference equation when, and only when, the associated parameter takes the smaller values of prime twins.

A guide to MATLAB: for beginners and experienced users , by Brian R. Hunt, Ronald L. Lipsman and Jonathan M. Rosenberg. Pp. 327. £22.95. 2001. ISBN 0 521 00859 X (Cambridge University Press).

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