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Waring problem with factorials

Published online by Cambridge University Press:  17 April 2009

Moubariz Z. Garaev ,
Florian Luca  and
Igor E. Shparlinski
Moubariz Z. Garaev
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México, e-mail: garaev@matmor.unam.mx, fluca@matmor.unam.mx
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México, e-mail: garaev@matmor.unam.mx, fluca@matmor.unam.mx
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
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We show that any residue class λ modulo p can be represented in the form n1! +…+ nℓ! ≡ λ (mod p) with ℓ = O((log p)3 log log p).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 2 , April 2005 , pp. 259 - 264
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Garaev, M.Z. and Luca, F., ‘Character sums and products of factorials modulo p’, J. Théor. Nombres Bordeaux (to appear).Google Scholar
[2]Garaev, M.Z., Luca, F. and Shparlinski, I.E., ‘Character sums and congruences with n!’, Trans. Amer. Math. Soc. 356 (2004), 5089–5102.CrossRefGoogle Scholar
[3]Garaev, M.Z., Luca, F. and Shparlinski, I.E., ‘Exponential sums and congruences with factorials’, J. Reine Angew. Math. (to appear).Google Scholar
[4]Guy, R.K., Unsolved problems in number theory (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
[5]Luca, F. and Shparlinski, I.E., ‘Prime divisors of shifted factorials’, Bull. Lond. Math. Soc. (to appear).Google Scholar
[6]Luca, F. and Shparlinski, I.E., ‘On the largest prime factor of n! + 2n − 1’, J. Théor. Nombresx Bordeaux (to appear).Google Scholar
[7]Luca, F. and Stănică, P., ‘Products of factorials modulo p’, Colloq. Math. 96 (2003), 191–205.CrossRefGoogle Scholar
[8]Stewart, C., ‘On the greatest and least prime factors of n! + 1, II’, Publ. Math. Debrecen 65 (2004), 461–480.CrossRefGoogle Scholar
[9]Vinogradov, I.M., Elements of number theory (Dover Publications, New York, 1954).Google Scholar