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A note on pseudoprimes with respect to abelian linear recurring sequence. (English) Zbl 0888.11006

It is proved that, for any finite system of simple abelian linear recurring sequences \(\{a_n^i\}\), \(i\in I\), and arbitrary integer \(l\geq 3\), Schinzel’s conjecture H implies the existence of infinitely many composite numbers \(n\) which are a product of \(l\) different primes and satisfy \(a_{ns}^i \equiv a_s^i \pmod n\) for every natural number \(s\).

MSC:

11A51 Factorization; primality
11B37 Recurrences
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References:

[1] ADAMS W.-SHANKS D.: Strong primality tests that are not sufficient. Math. Comp. 39 (1982), 255-300. · Zbl 0492.10005 · doi:10.2307/2007637
[2] JAKUBEC S.-NEMOGA K.: On a conjecture concerning sequences of the third order. Math. Slovaca 36 (1986), 85-89. · Zbl 0583.10007
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[4] MARKO F.: Pseudoprimes with Respect to Linear Recurring Sequences. (Slovak). Thesis, SAV, Bratislava, 1991.
[5] MILLER J. O P.-SPENCER BROWN G., SPENCER BROWN D. J.: The identification of prime numbers.
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