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Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007519(n), n>=1 (primes congruent to 1 mod 8).
+10
12
5, 7, 9, 11, 13, 11, 13, 15, 19, 21, 17, 17, 21, 25, 19, 23, 21, 21, 29, 23, 23, 31, 33, 25, 27, 25, 29, 31, 31, 29, 29, 37, 41, 31, 35, 31, 37, 39, 41, 43, 35, 39, 35, 35, 43, 35, 49, 41, 37
COMMENTS
For the corresponding term y1(n) see 2* A254761(n). For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254762(n) and 2* A254763(n). The present solutions of this first class are the smallest positive ones.
See the Nagell reference Theorem 111, p. 210, for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208, for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable and the primes from A007519 do not divide 4. The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n). The intervals to be scanned are ceiling((sqrt(8 + p(n))-1)/2) <= X1(n) <= floor((sqrt(2*p(n))-1)/2), with p(n) = A007519(n), and 1 <= Y1(n) <= floor(sqrt( A005123(n))). The general positive proper solutions are for both classes obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the positive fundamental column vectors (x(n),y(n))^T. The n-th power M^n = S(n-1, 6)*M - S(n-2, 6) 1_2 , where 1_2 is the 2 X 2 identity matrix and S(n, 6), with S(-2, 6) = -1 and S(-1, 6) = 0 is the Chebyshev S-polynomial evaluated at x = 6, given in A001109(n). The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including in the first class also the prime 2) are given in A002334. - Wolfdieter Lang, Feb 12 2015
REFERENCES
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
FORMULA
a(n)^2 - 2*(2* A254760(n))^2 = A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007519(n) as first entry): [17, [5, 2]], [41, [7, 2]], [73, [9, 2]], [89, [11, 4]], [97, [13, 6]], [113, [11, 2]], [137, [13, 4]], [193, [15, 4]], [233, [19, 8]], [241, [21, 10]], [257, [17, 4]], [281, [17, 2]], [313, [21, 8]], [337, [25, 12]], [353, [19, 2]], [401, [23, 8]], [409, [21, 4]], ...
n=1: 5^2 - 2*2^2 = 25 - 8 = 17, ...
One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007519(n), n>=1 (primes congruent to 1 mod 8).
+10
8
2, 4, 6, 5, 4, 8, 7, 9, 7, 6, 11, 14, 9, 7, 16, 11, 15, 18, 8, 14, 20, 10, 9, 19, 15, 22, 14, 13, 16, 20, 23, 13, 11, 25, 17, 28, 16, 15, 14, 13, 23, 18, 26, 29, 16, 32, 13, 20, 28, 24
COMMENTS
The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A254762(n). See the comments and the Nagell reference in A254760.
FORMULA
A254762(n)^2 - 2*(2*a(n))^2 = A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
n = 2: 13^2 - 2*(2*4)^2 = 169 - 128 = 41.
The smallest positive solution is (x1(2), y1(2)) = (7, 2) from ( A254760(2), 2* A254761(2)). a(4) = 11 - 3*2 = 5.
One half of the fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007519(n), n >= 1 (primes congruent to 1 mod 8).
+10
7
1, 1, 1, 2, 3, 1, 2, 2, 4, 5, 2, 1, 4, 6, 1, 4, 2, 1, 7, 3, 1, 7, 8, 2, 4, 1, 5, 6, 5, 3, 2, 8, 10, 2, 6, 1, 7, 8, 9, 10, 4, 7, 3, 2, 9, 1, 12, 7, 3, 5
COMMENTS
For the corresponding term x1(n) see A254760(n). See A254760 also for the Nagell reference. The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including in the first class also prime 2) are given in A002335.
FORMULA
A254760(n)^2 - 2*(2*a(n))^2 = A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
n = 3: 9^2 - 2*(2*1)^2 = 81 - 8 = 73.
Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).
+10
4
5, 11, 9, 17, 13, 23, 21, 17, 27, 35, 23, 21, 41, 31, 29, 39, 37, 53, 33, 31, 41, 59, 39, 49, 37, 35, 43, 63, 53, 37, 49, 77, 59, 47, 75, 83, 65, 53, 73, 51, 45, 61, 71, 59, 79, 69, 95, 55, 49, 101
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 = A007522(n) = 7 + 8* A139487(n) is given in A254929(n). The positive fundamental solutions of the first classes are given in ( A254764(n), A254765(n)). For comments and the Nagell reference see A254764.
FORMULA
a(n)^2 - 2*( A254929(n))^2 = A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
The first pairs [x2(n), y2(n)] of the fundamental positive solutions of the second class are (we list the prime A007522(n) as first entry): [7,[5,3]], [23,[11,7]], [31,[9,5]], [47,[17,11]], [71,[13,7]], [79,[23,15]], [103,[21,13]], [127,[17,9]], [151,[27,17]], [167,[35,23]], [191,[23,13]], [199,[21,11]], [223,[41,27]], [239,[31,19]], [263,[29,17]], [271,[39,25]], ...
Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007522(n), n>=1 (primes congruent to 7 mod 8).
+10
4
3, 7, 5, 11, 7, 15, 13, 9, 17, 23, 13, 11, 27, 19, 17, 25, 23, 35, 19, 17, 25, 39, 23, 31, 21, 19, 25, 41, 33, 19, 29, 51, 37, 27, 49, 55, 41, 31, 47, 29, 23, 37, 45, 35, 51, 43, 63, 31, 25, 67
COMMENTS
The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A254766(n). See the comments and the Nagell reference in A254764.
FORMULA
A254766(n)^2 - 2*a(n)^2 = A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
n = 2: 11^2 - 2*7^2 = 121 - 98 = 23.
The smallest positive solution is (x1(2), y1(2)) = (5, 1) from ( A254764(2), A254765(2)). a(4) = 2*7 - 3*1 = 11.
Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).
+10
3
3, 5, 7, 7, 11, 9, 11, 15, 13, 13, 17, 19, 15, 17, 19, 17, 19, 19, 23, 25, 23, 21, 25, 23, 27, 29, 29, 25, 27, 35, 31, 27, 29, 33, 29, 29, 31, 35, 31, 37, 43, 35, 33, 37, 33, 35, 33, 41, 47, 35
COMMENTS
For the corresponding term y1(n) see A254765(n). For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254766(n) and A254929(n). The present solutions of the first class are the smallest positive ones.
See the Nagell reference Theorem 111 p. 210 for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208 for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable, and the primes A007522(n) do not divide 4. The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) + 1. The intervals for X1(n) and Y1(n) to be scanned are ceiling((sqrt(2+p(n))-1)/2) <= X1(n) <= floor(sqrt((2*p(n))-1)/2), with p(n) = A007522(n) and 0 <= Y1(n) <= floor((sqrt(p(n)/2)-1)/2). The general positive proper solutions for both classes are obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental column vectors (x(n),y(m))^T.
The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002334.
REFERENCES
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
FORMULA
a(n)^2 - 2* A254765(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.
EXAMPLE
The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007522(n) as first entry): [7, [3, 1]], [23, [5, 1]], [31, [7, 3]], [47, [7, 1]], [71, [11, 5]], [79, [9, 1]], [103, [11, 3]], [127, [15, 7]], [151, [13, 3]], [167, [13, 1]], [191, [17, 7]], [199, [19, 9]], [223, [15, 1]], ...
Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).
+10
3
1, 1, 3, 1, 5, 1, 3, 7, 3, 1, 7, 9, 1, 5, 7, 3, 5, 1, 9, 11, 7, 1, 9, 5, 11, 13, 11, 3, 7, 17, 11, 1, 7, 13, 3, 1, 7, 13, 5, 15, 21, 11, 7, 13, 5, 9, 1, 17, 23, 1
COMMENTS
For the corresponding term x1(n) see A254764(n). See A254764 for comments and the Nagell reference. The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002335.
FORMULA
A254764(n)^2 - 2*a(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.
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).
+10
3
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
COMMENTS
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.
FORMULA
a(n)^2 - 2* A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.
EXAMPLE
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
...
MATHEMATICA
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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