1 |
H. Goral and D. C. Sertbas, Euler sums and non-integerness of harmonic type sums, Hacet. J. Math. Stat. 49 (2020), no. 2, 586-598. https://doi.org/10.15672/hujms.544489
DOI
|
2 |
D. R. Heath-Brown, The differences between consecutive primes. III, J. London Math. Soc. (2) 20 (1979), no. 2, 177-178. https://doi.org/10.1112/jlms/s2-20.2.177
DOI
|
3 |
I. Mezo, About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 50 (2007), 13-20 (2009).
|
4 |
G. Faltings, Endlichkeitss¨atze f¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern, Invent. Math. 73 (1983), no. 3, 349-366. https://doi.org/10.1007/BF01388432
DOI
|
5 |
E. Alkan, H. Goral, and D. C. Sertbas, Hyperharmonic numbers can rarely be integers, Integers 18 (2018), Paper No. A43, 15 pp.
|
6 |
R. A. Amrane and H. Belbachir, Are the hyperharmonics integral? A partial answer via the small intervals containing primes, C. R. Math. Acad. Sci. Paris 349 (2011), no. 3-4, 115-117. https://doi.org/10.1016/j.crma.2010.12.015
DOI
|
7 |
T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.
|
8 |
Y. F. Bilu and R. F. Tichy, The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), no. 3, 261-288. https://doi.org/10.4064/aa-95-3-261-288
DOI
|
9 |
J. H. Conway and R. K. Guy, The book of Numbers, Copernicus, New York, 1996. https://doi.org/10.1007/978-1-4612-4072-3
DOI
|
10 |
SageMath, the Sage Mathematics Software System (Version 8.3), (2018). The Sage Developers, http://www.sagemath.org
|
11 |
W. Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
|
12 |
H. Goral and D. C. Sertba,s, Almost all hyperharmonic numbers are not integers, J. Number Theory 171 (2017), 495-526. https://doi.org/10.1016/j.jnt.2016.07.023
DOI
|
13 |
R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532-562. https://doi.org/10.1112/plms/83.3.532
DOI
|
14 |
D. C. Sertba,s, Harmonic type sums and their arithmetic properties, PhD thesis, Koc University, 2020.
|
15 |
J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. https://doi.org/10.1007/978-1-4757-4252-7
DOI
|
16 |
R. A. Amrane and H. Belbachir, Non-integerness of class of hyperharmonic numbers, Ann. Math. Inform. 37 (2010), 7-11.
|
17 |
H. Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), no. 1, 23-46.
DOI
|
18 |
D. C. Sertbas, Hyperharmonic integers exist, C. R. Math. Acad. Sci. Paris 358 (2020), no. 11-12, 1179-1185. https://doi.org/10.5802/crmath.137
DOI
|
19 |
R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 65, Longman Scientific & Technical, Harlow, 1993.
|
20 |
A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), no. 6, 87-105.
|
21 |
C. L. Siegel, Uber einige Anwendungen diophantischer Approximationen, Abhandlungen der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse Nr. 1, 1929.
|
22 |
L. Theisinger, Bemerkung uber die harmonische Reihe, Monatsh. Math. Phys. 26 (1915), no. 1, 132-134. https://doi.org/10.1007/BF01999444
DOI
|
23 |
J. Kurschak, On the harmonic series, Matematikaies Fizikai Lapok 27 (1918), 299-300.
|
24 |
M. Hindry and J. H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1210-2
DOI
|