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A254930
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Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).
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3
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5, 7, 11, 9, 13, 17, 13, 19, 23, 17, 15, 21, 25, 17, 23, 27, 35, 23, 29, 21, 41, 25, 31, 23, 35, 29, 39, 43, 37, 31, 27, 49, 53, 33, 31, 37, 47, 41, 55, 59, 31, 45, 39, 49, 37, 35, 61, 37, 35
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OFFSET
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1,1
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COMMENTS
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The corresponding terms y = y2(n) are given in A254931(n).
There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).
The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.
The second class solutions give the second smallest positive integer solutions of this Pell equation.
For comments and the Nagell reference see A254760.
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LINKS
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FORMULA
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a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.
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EXAMPLE
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The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
1 7 5 3
2 17 * 7 4
3 23 11 7
4 31 9 5
5 41 * 13 8
6 47 17 11
7 71 13 7
8 73 * 19 12
9 89 * 17 10
10 97 * 15 8
11 103 21 13
12 113 * 25 16
13 127 17 9
14 137 * 23 14
15 151 27 17
16 167 35 23
17 191 23 13
18 193 * 29 18
19 199 21 11
20 223 41 27
...
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MATHEMATICA
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Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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