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**1 - 3**of**3**### Solving the generalized Pell equation x 2 − Dy 2 = N

"... This article gives fast, simple algorithms to find integer solutions x, y to generalized Pell equations, x 2 − Dy 2 = N, for D a positive integer, not a square, and N a nonzero integer. Pell equations have fascinated for centuries. ..."

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This article gives fast, simple algorithms to find integer solutions x, y to generalized Pell equations, x 2 − Dy 2 = N, for D a positive integer, not a square, and N a nonzero integer. Pell equations have fascinated for centuries.

### NOTES ON SOME NEW KINDS OF PSEUDOPRIMES

"... Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where ..."

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Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2-pseudoprimes to the same bases. In Section 3, we give examples of Sylow p-pseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow p-pseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a k-fold Carmichael Sylow pseudoprime, ifitisaSylowp-pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3-fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Syl-upsp(b1,...,bk) for short, if it is a Sylp-psp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Syl-upsp(2, 3, 5)’s < 1016 and some Syl-upsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.

### On p x − q y = c and related three term exponential Diophantine equations with prime bases short running title: Prime Base Exponential Diophantine Equations

, 2011

"... Using a theorem on linear forms in logarithms, we show that the equation p x − 2 y = p u − 2 v has no solutions (p, x, y, u, v) withx̸ = u, wherepis a positive prime and x, y, u, andvare positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 1015. More gener ..."

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Using a theorem on linear forms in logarithms, we show that the equation p x − 2 y = p u − 2 v has no solutions (p, x, y, u, v) withx̸ = u, wherepis a positive prime and x, y, u, andvare positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 1015. More generally, we obtain a similar result for px − qy = pu − qv> 0whereqis a positive prime, q ̸ ≡ 1 mod 12. We solve a question of Edgar showing there is at most one solution (x, y) topx − qy =2h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x, y) to|px ± qy | = c and at most two solutions (x, y, z) topx ± qy ± 2z =0,for given positive primes p and q and integer c.