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A254936
Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).
5
9, 11, 13, 19, 25, 15, 21, 23, 35, 41, 25, 21, 37, 49, 23, 39, 29, 25, 57, 35, 27, 59, 65, 33, 43, 29, 49, 55, 51, 41, 37, 69, 81, 39, 59, 35, 65, 71, 77, 83, 51, 67, 47, 43, 79, 39, 97, 69, 49, 59
OFFSET
1,1
COMMENTS
The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007519(n) = -(1 + 8*A005123(n)) is given in A254937(n).
For comments and the Nagell reference see A254934.
FORMULA
a(n)^2 - 2*A254937(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = -(3*A254934(n) - 4*A254935(n)), n >= 1.
EXAMPLE
The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007519(n) appears as first entry):
[17, [9, 7]], [41, [11, 9]], [73, [13, 11]],
[89, [19, 15]], [97, [25, 19]], [113, [15, 13]],
[137, [21, 17]], [193, [23, 19]], [233, [35, 27]],
[241, [41, 31]], [257, [25, 21]], [281, [21, 19]],
[313, [37, 29]], [337, [49, 37]], [353, [23, 21]],
[401, [39, 31]], [409, [29, 25]], [433, [25, 23]],
[449, [57, 43]], [457, [35, 29]], [521, [27, 25]],
[569, [59, 45]], [577, [65, 49]], [593, [33, 29]],
[601, [43, 35]], [617, [29, 27]], [641, [49, 39]], ...
a(4) = -(3*3 - 4*7) = 28 - 9 = 19.
KEYWORD
nonn,look,easy
AUTHOR
Wolfdieter Lang, Feb 18 2015
STATUS
approved