Finiteness results for Diophantine triples with repdigit values

Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb N$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples $(a,b,c)\in \mathbb N^3$ with $c<b<a$ such that $ab+1,ac+1$ and $ab+1$ lie in the set $\mathcal R_g$. In this paper, we prove effective finitness results for the set $\mathcal D_g$

Various Arithmetic Functions and their Applications

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): The Florentin Smarandache papers special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, România). The book is based on various articles in the theory of numbers (starting from 1975), updated many times. Special thanks to C. Dumitrescu and V. Seleacufrom the University of Craiova (see their edited book Some Notions and Questions in Number Theory , Erhus Press, Glendale, 1994), M. Bencze, L. Tutescu, E. Burton, M. Coman, F. Russo, H. Ibstedt, C. Ashbacher, S. M. Ruiz, J. Sandor, G. Policarp, V. Iovan, N. Ivaschescu, etc. who helped incollecting and editing this material. This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume Solving Diophantine Equations , published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira – with his algorithmic thinking and knowledge of Mathcad