Abstract
We obtain a lower bound on the number of prime divisors of integers whose g-ary expansion contains a fixed number of nonzero digits.
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REFERENCES
M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Preprint, 2002, 1–9.
C. Ballot, Density of prime divisors of linear recurrences, Memoirs Amer. Math. Soc., 115, Amer. Math. Soc., Providence, RI, 1995.
R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), 331–361.
P. Erdős and R. Murty, On the order of a (mod p), Proc. 5 th Canadian Number Theory Association Conf., Amer. Math. Soc., Providence, RI, 1999, 87–97.
E. Fouvry and C. Mauduit, Méthodes des crible et fonctions sommes des chiffres, Acta Arith. 77 (1996), 339–351.
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann. 305 (1996), 571–599.
E. Fouvry, Théorème de Brun–Titchmarsh: Application au théorème de Fermat, Invent. Math. 79 (1985), 383–407.
H. Hasse, Ñber die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a ≠ 0 von gerader bzw. ungerader Ordnung mod p ist, Math. Ann. 166 (1966), 19–23.
D. R. Heath-Brown and S. V. Konyagin, New bounds for Gauss sums derived from k th powers, and for Heilbronn's exponential sum, Ouart. J. Math. 51 (2000), 221–235.
H.-K. Indlekofer and N. M. Timofeev, Divisors of shifted primes, Publ. Math. Debrecen 60 (2002), 307–345.
S. Konyagin, Arithmetic properties of integers with missing digits: distribution in residue classes, Periodica Math. Hungar. 42 (2001), 145–162.
S. V. Konyagin, C. Mauduit and A. Sárközy, On the number of prime factors of integers characterized by digit properties, Period. Math. Hungarica 40 (2000), 37–52.
S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 1999.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math. 118 (1985), 449–461.
F. Luca, Arithmetic properties of positive integers with fixed digit sum, Preprint, 2002.
F. Luca, Arithmetic properties of members of a binary recurrent sequence, Acta Arith. (to appear).
C. Mauduit and A. Sárközy, On the arithmetic structure of sets characterized by sum of digits properties,J. Number Theory 61 (1996), 25–38.
C. Mauduit and A. Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith. 81 (1997), 145–173.
P. Moree, Counting divisors of Lucas numbers, Pacific J. Math. 186 (1998), 267–284.
P. Moree and P. Stevenhagen, Prime divisors of the Lagarias sequence, J. Théor. Nombres Bordeaux 13 (2001), 241–251.
R. Murty and S. Wong, The ABC conjecture and prime divisors of the Lucas and Lehmer sequences, Preprint, 2001.
F. Pappalardi, On the order of finitely generated subgroups of ?* (mod p) and divisors of p- 1, J. Number Theory 57 (1996), 207–222.
T. N. Shorey and C. L. Stewart, On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers, II, J. London Math. Soc. 23 (1981), 17–23.
I. E. Shparlinski, On the distribution of the Diffie–Hellman pairs, Finite Fields and Their Appl. 8 (2002), 131–141.
I. E. Shparlinski, Cryptographic applications of analytic number theory: Complexity lower bounds and pseudorandomness, Birkhäuser, 2002.
C. L. Stewart, On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers, Proc. London Math. Soc. 35 (1977), 425–447.
C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math. 319 (1980), 63–72.
I. M. Vinogradov, Elements of number theory, Dover Publ., New York, 1954.
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Shparlinski, I.E. Prime divisors of sparse integers. Periodica Mathematica Hungarica 46, 215–222 (2003). https://doi.org/10.1023/A:1025996312037
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DOI: https://doi.org/10.1023/A:1025996312037