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A339519
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Odd composite integers m such that A087130(2*m-J(m,29)) == 5*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.
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3
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9, 15, 27, 39, 45, 91, 117, 121, 135, 143, 195, 287, 351, 507, 585, 741, 1521, 1547, 1573, 1755, 1935, 2015, 2067, 2535, 2601, 3157, 3227, 3445, 3505, 3519, 3731, 4563, 4879, 4921, 6201, 6273, 6543, 6591, 6721, 7605, 7803, 8099, 10335, 10377, 10403, 10515
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OFFSET
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1,1
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COMMENTS
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The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=5, D=29 and k=2, while V(m) recovers A087130(m).
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REFERENCES
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D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
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LINKS
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MATHEMATICA
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Select[Range[3, 20000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 29], 5] - 5*JacobiSymbol[#, 29], #] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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