There are finitely many even perfect polynomials over $F_p$ with p + 1 irreducible divisors
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Abstract
We give, for a fixed prime number p, necessary conditions for the existence of products of p+1 distinct monic irreducible polynomials in one variable over the finite field F_p, each raised to an arbitrary positive power, that are even perfect polynomials; i.e., they have at least one root in F_p, and they are equal to the sum of all their monic divisors.As a consequence, we prove that the set of such polynomials is finite, and if q = (p−1)/2 is also prime, so that q is a Sophie Germain prime and p is a safe prime, then it is empty. This is the first known finiteness result for perfect polynomials. We might consider it as an analogue of Dickson’s result that proves the finiteness of the set of odd perfect numbers with a fixed number of distinct prime divisors, each raised to an arbitrary positive power.
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Gallardo, L., & Rahavandrainy, O.
(2016).
There are finitely many even perfect polynomials over $F_p$ with p + 1 irreducible divisors.
Acta Mathematica Universitatis Comenianae, 85(2), 261-275.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/241/389
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References
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polynomials over GF(q), Rend. Accad. Lincei 62 (1977), 283–291.
[2] J. T. B. Beard Jr., Perfect polynomials revisited, Publ. Math. Debrecen 38, No. 1/2 (1991), 5–12.
[3] C. Berge, Principles of Combinatorics, New York (1971).
[4] E. F. Canaday, The sum of the divisors of a polynomial, Duke Math. Journal 8 (1941), 721–737.
[5] L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American J. 35 (1913), 413–422.
[6] H. Dubner, Large Sophie Germain primes, Math. Comp. 65 - No. 213 (1996), 393–396.
[7] L. Gallardo, O. Rahavandrainy, Perfect polynomials over F4 with less than five prime factors, Portugaliae Mathematica 64 - Fasc. 1 (2007), 21–38.
[8] L. H. Gallardo, O. Rahavandrainy, Odd perfect polynomials over F2, J. Th´eor. Nombres Bordeaux 19 (2007), 167–176.
[9] L. H. Gallardo, O. Rahavandrainy, On perfect polynomials over Fp with p irreducible factors, Portugaliae Mathematica 69 - Fasc. 4 (2012), 283–303.
[10] L. H. Gallardo, O. Rahavandrainy, Perfect polynomials over Fp with p + 1 irreducible divisors, AMUC Journal LXXXIII, 1 (2014), 93–112.
[11] K. H. Indlekofer, A. J´arai, Largest known twin primes and Sophie Germain primes, Math. Comp. 68 - No. 227 (1999), 1317–1324.
[12] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its applications, Cambridge University Press, 1983 (Reprinted 1987).
[13] D. G. Mead, Newton’s Identities, Amer. Math. Monthly 99 - No. 8 (1992), 749–751.
[14] M. A. Nyblom, Sophie Germain primes and the exceptional values of the equal-sum-and-product problem, Fibonacci Quart. 50 - No. 1 (2012), 58–61.
[15] OEIS, On-line Encyclopedia of Integer Sequences, https://oeis.org. [16] J. P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 - No. 4 (2003), 429–440.
[17] S. Yates, Sophie Germain primes, The mathematical heritage of C. F. Gauss, World Sci. Publ., River Edge, NJ (1991), 882–886.