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Number Theory in the Spirit of Liouville

Cambridge University Press
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Joseph Liouville (1809–1882) was a hard-working French mathematician. He is, or should be, remembered for his theorem that a bounded entire function is a constant (from which the Fundamental Theorem of Algebra follows), for his construction of the first class of transcendental numbers, for some of the first theorems on integration in finite terms, and for Sturm-Liouville differential equations. In addition, in 1836 he founded the Journal des mathématiques pures et appliquées in which he published the works of Galois in 1846, Liouville having been the first to notice their value.

He did much other work in the time left over from scrambling for positions, teaching thirty or more hours a week, and dealing with enemies and ill health. In 1879, Thomas Hirst wrote in his diary, “A little shrivelled gouty old man, [Liouville] has become very garrulous. It was with difficulty I broke away from him.” His old age was unfortunately marked with bitterness. A crater on the moon is named for him.

When he was forty-seven, Liouville became struck with number theory and devised a new method. From 1858 to 1865 he published eighteen papers on it, and he applied it in ninety others. It is entirely elementary (though not simple) and with its aid he was able to prove many new results and to derive many others that had been obtained with more advanced methods. The first of his formulas is that if f is an even function, then the sum of f (a + b) − f (ab), taken over odd integers a, b, x, y such that ax + by = 2n, is equal to the sum of m(f (2m) − f (0)) taken over the integers m that divide n such that n/m is odd. It is to be expected that a formula connecting an ordinary sum with one over the divisors of an integer would produce number-theoretical results, and so it does. One of the many to be found in the book is that the number of ways of representing n as a sum of four triangular numbers is the sum of the divisors of 2n + 1.

As Professor Williams points out, though Liouville’s ideas are 150 years old they do not appear in elementary number theory courses and there is no book in English devoted to them. He aims “to remedy this situation by providing a gentle introduction to Liouville’s method.” This he has done. He gives some of Liouville’s identities and applies them to an amazing variety of situations. The notation is dense — some of the summation signs have four lines of conditions under them — but the writing is clear and easily followed. There are exercises at the end of nineteen chapters, though without hints or solutions. The list of references contains two hundred and seventy-four entries.

This book will not be superseded anytime soon, if ever. It is the place to go for anyone who is curious about Liouville’s method.

Woody Dudley’s claim to number-theoretical fame is that his elementary number theory text is still in print, forty-two years after it was first published.

Date Received: 
Monday, December 13, 2010
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Kenneth S. Williams
London Mathematical Society Student Texts 76
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Underwood Dudley
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1. Joseph Liouville (1809–1888)
2. Liouville's ideas in number theory
3. The arithmetic functions σk(n), σk*(n), dk,m(n) and Fk(n)
4. The equation i2 + jk = n
5. An identity of Liouville
6. A recurrence relation for σ*(n)
7. The Girard–Fermat theorem
8. A second identity of Liouville
9. Sums of two, four and six squares
10. A third identity of Liouville
11. Jacobi's four squares formula
12. Besge's formula
13. An identity of Huard, Ou, Spearman and Williams
14. Four elementary arithmetic formulae
15. Some twisted convolution sums
16. Sums of two, four, six and eight triangular numbers
17. Sums of integers of the form x2+xy+y2
18. Representations by x2+y2+z2+2t2, x2+y2+2z2+2t2 and x2+2y2+2z2+2t2
19. Sums of eight and twelve squares
20. Concluding remarks

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Saturday, May 14, 2011