Geodesic
A~Diophantine Representation of Bernoulli Numbers and Its Applications
Informatics and Automation, Mathematical logic and algebra, Tome 242 (2003), pp. 98-102.

Voir la notice de l'article provenant de la source Math-Net.Ru

A new method for constructing a Diophantine representation of Bernoulli numbers is proposed. The method is based on the Taylor series for the function $\tau /(e^\tau -1)$. This representation can be used for constructing Diophantine representations of the set of all Carmichael numbers (i.e. numbers that are pseudoprime for every base) and for the set of all square-free numbers. 
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Yu. V. Matiyasevich. A~Diophantine Representation of Bernoulli Numbers and Its Applications. Informatics and Automation, Mathematical logic and algebra, Tome 242 (2003), pp. 98-102. http://geodesic.mathdoc.fr/item/TRSPY_2003_242_a6/

[1] M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR

[2] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1985 | MR | Zbl

[3] Vsemirnov M. A., “Beskonechnye mnozhestva prostykh chisel, dopuskayuschie diofantovy predstavleniya s vosemyu peremennymi”, Zap. nauch. seminarov POMI, 220, 1995, 36–48 | MR | Zbl

[4] Jones J. P., “Diophantine representation of Mersenne and Fermat primes”, Acta Arith., 35:3 (1979), 209–221 | MR | Zbl

[5] Jones J. P., “Universal Diophantine equation”, J. Symb. Logic, 47:3 (1982), 549–571 | DOI | MR | Zbl

[6] Jones J. P., Sato D., Wada H., Wiens D., “Diophantine representation of the set of prime numbers”, Amer. Math. Month., 83:6 (1976), 449–464 | DOI | MR | Zbl

[7] Koblitz N., A course in number theory and cryptography, Grad. Texts Math., 114, Springer, New York etc., 1987 | MR | Zbl

[8] Matiyasevich Yu. V., “Diofantovost perechislimykh mnozhestv”, DAN SSSR, 191:2 (1970), 279–282 | Zbl

[9] Matiyasevich Yu. V., “Diofantovo predstavlenie mnozhestva prostykh chisel”, DAN SSSR, 196:4 (1971), 770–773 | Zbl

[10] Matiyasevich Yu., “Prostye chisla perechislyayutsya polinomom ot 10 peremennykh”, Zap. nauch. seminarov LOMI, 68, 1977, 62–82 ; J. Sov. Math., 15:1 (1981), 33–44 | Zbl | DOI | MR | Zbl

[11] Matijasevič Yu. V., “Some purely mathematical results inspired by mathematical logic”, Logic, foundations of mathematics, and computability theory, Proc. Fifth Intern. Congr. of Logic, Methodology and Philosophy of Science, v. 1 (London, Ontario, Canada, 1975), eds. R. E. Butts, J. Hintikka, D. Reidel, Dordrecht, 1977, 121–127 | MR

[12] Matiyasevich Yu., Desyataya problema Gilberta, Fizmatlit, M., 1993 | MR

[13] Putnam H., “An unsolvable problem in number theory”, J. Symb. Logic, 25:3 (1960), 220–232 | DOI | MR

[14] Robinson J., “Unsolvable Diophantine problems”, Proc. Amer. Math. Soc., 22:2 (1969), 534–538 | DOI | MR | Zbl