Geodesic
A~Diophantine Representation of Bernoulli Numbers and Its Applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 98-102

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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. 
@article{TM_2003_242_a6,
     author = {Yu. V. Matiyasevich},
     title = {A~Diophantine {Representation} of {Bernoulli} {Numbers} and {Its} {Applications}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {98--102},
     publisher = {mathdoc},
     volume = {242},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_242_a6/}
}
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Yu. V. Matiyasevich. A~Diophantine Representation of Bernoulli Numbers and Its Applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 98-102. http://geodesic.mathdoc.fr/item/TM_2003_242_a6/