## Tuesday, October 20, 2009

### The Axiom of Choice

Three useful links dealing with the Axiom of Choice: The Axiom of Choice - Stanford Encyclopedia of Philosophy entry by John L. Bell, the University of Western Ontario; The Axiom of Choice (a short paper by Prof. John L. Bell) ; The Relative Consistency of the Axiom of Choice - Mechanized Using Isabelle/ZF by Lawrence C. Paulson, Computer Laboratory Univ. of Cambridge. An important result, in a categorical setting linked to intutionistic set theory, was obtained by Radu Diaconescu (Axiom of choice and complementation, Proc. Amer. Math. Soc. 51, 1975, 176-178): a topos satisfying the axiom of choice must be boolean (in short... the axiom of choice implies the law of the excluded middle).

## Wednesday, October 14, 2009

## Tuesday, October 13, 2009

### A conjecture on primes

The Feit-Thompson conjecture: there are no distinct prime numbers p and q for which (p^q-1)/(p-1) divides (q^p-1)/(q-1). Note that a stronger statement stating that (p^q-1)/(p-1) and (q^p-1)/(q-1) are relatively prime whenever p and q are distinct primes does not hold.

Indeed, 112643 = GCD((3313^17 -1)/3312, (17^3313-1)/16) - see Stevens (1971).

Note that (3313^17 - 1)/3312 factors as

78115430278873040084455537747447422887 * 23946003637421 * 112643,

while (17^3313-1)/16 modulo (3313^17 -1)/3312 equals...

149073454345008273252753518779212742886488244343395482423

The 1970 Fields medalist and 2008 Abel prize winner John Griggs Thompson was born on Ottawa, KS on October 13.

Indeed, 112643 = GCD((3313^17 -1)/3312, (17^3313-1)/16) - see Stevens (1971).

Note that (3313^17 - 1)/3312 factors as

78115430278873040084455537747447422887 * 23946003637421 * 112643,

while (17^3313-1)/16 modulo (3313^17 -1)/3312 equals...

149073454345008273252753518779212742886488244343395482423

The 1970 Fields medalist and 2008 Abel prize winner John Griggs Thompson was born on Ottawa, KS on October 13.

## Sunday, October 11, 2009

### Ferdinand Eisenstein (16 April 1823 – 11 October 1852)

Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852)

Eisenstein's Mac Tutor Biography

The life of Gotthold Ferdinand Eisenstein - by M. Schmitz (Res. Lett. Inf. Math. Sci., 2004, Vol. 6, pp 1-13)

Eisenstein series

Eisenstein Snowflakes (YouTube)

Eisenstein's Mac Tutor Biography

The life of Gotthold Ferdinand Eisenstein - by M. Schmitz (Res. Lett. Inf. Math. Sci., 2004, Vol. 6, pp 1-13)

Eisenstein series

Eisenstein Snowflakes (YouTube)

## Sunday, October 4, 2009

### Dimitrie Pompeiu (1873–1954)

Dimitrie Pompeiu (October 4, 1873, Broscǎuţi, Botoşani – October 8, 1954, Bucharest) - orthodox christian, and one of the greatest Romanian mathematicians. He is remembered in the mathematical world for numerous contributions such as: the set distance [1] that he introduced in his 1905 Université de Paris dissertation published in the same year in the

REFERENCES

[1] T. Bârsan and D. Tiba. One hundred years since the introduction of the set distance by Dimitrie Pompeiu. Institute of Mathematics of the Romanian Academy.

[2] D. Pompeiu. Sur une classe de fonctions d'une variable complexe. Rendiconti del Circolo Matematico di Palermo, t. XXXIII, Ist sem. 1912, pp. 108-113.

[3] Pompeiu's biography from the The MacTutor History of Mathematics archive.

[4] D. Pompeiu.

[5] R. Remmert. Theory of Complex Functions. Graduate Texts in Mathematics, Springer Verlag, 2nd Edition (1989).

[6] D. Zeilberger. Pompeiu's problem on Discrete Space. Proc. Natl. Acad. Sci. USA, Vol. 75 (8), 3555-3556 (1978).

*Annales de la faculté des sciences de Toulouse*(a set distance was introduced in a slightly different form in 1914 by Hausdorff, who credited Pompeiu's definition though), his contributions in complex analysis, including the areolar derivative [2] and the seminal Cauchy-Pompeiu's formula (higher dimensional analogues of the Cauchy-Pompeiu formula are topics of current research, while the formula was used in the theory of functions of several complex variables by Dolbeault and Grothendieck [5]), and for the celebrated Pompeiu's Conjecture that he formulated in his 1929 C. R. Acad. Sci. Paris article [4], a conjecture not fuly proved yet. Still, elegant analogues of Pompeiu's Conjecture continue to be proved in other areas [6] - this is an indicator of the fertility of the idea.REFERENCES

[1] T. Bârsan and D. Tiba. One hundred years since the introduction of the set distance by Dimitrie Pompeiu. Institute of Mathematics of the Romanian Academy.

[2] D. Pompeiu. Sur une classe de fonctions d'une variable complexe. Rendiconti del Circolo Matematico di Palermo, t. XXXIII, Ist sem. 1912, pp. 108-113.

[3] Pompeiu's biography from the The MacTutor History of Mathematics archive.

[4] D. Pompeiu.

*Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables*, Comptes Rendus de l'Académie des Sciences Paris Série I. Mathématique, 188, 1138 –1139 (1929).[5] R. Remmert. Theory of Complex Functions. Graduate Texts in Mathematics, Springer Verlag, 2nd Edition (1989).

[6] D. Zeilberger. Pompeiu's problem on Discrete Space. Proc. Natl. Acad. Sci. USA, Vol. 75 (8), 3555-3556 (1978).

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