A329587 - OEIS

%I #25 Mar 14 2021 14:23:46

%S 3,2,1,2,4,3,2,3,5,4,7,4,1,4,3,4,5,2,6,5,4,5,5,8,7,6,11,6,1,6,5,6,8,7,

%T 6,7,5,3,10,3,5,12,10,12,9,8,15,8,1,8,7,8,10,9,8,9,5,2,11,10,15,2,15,

%U 8,15,12,19,10,1,10,5,8,5,12,5,18,9,10,15,18,12,11,10,11,13,12,17,12,19,12,23,12,1,12,5,12,7,12,11,12

%N Irregular triangle read by rows: representative solutions (a, b) of the complex congruence z^2 == +1 (mod m), where z = a + b*i = r*exp(i*phi), with nonvanishing a*b, for all positive integers m which have solutions.

%C The length of row n is given in 2*A329588(n).

%C This table should be considered together with A329585 which gives the solutions of this congruence with a*b = 0 (real or pure imaginary solutions) for each m >= 2. Only for m = 1 is a = b, namely (a, b) = (0, 0).

%C Because z^2 = (a^2 - b^2) + 2*a*b*i, this is equivalent to the two congruences (i) a^2 - b^2 == 1 (mod m) and (ii) 2*a*b == 0 (mod m). Here with nonvanishing a*b.

%C Because with each solution z = a + b*i, with representative a and b from {1, 2, ..., m-1}, also -z = -(a + b*i), zbar:= a - b*i and -zbar = -a + b*i are solutions modulo m.

%C This table T(n, k) lists all pairs (a,b) for those even m = m(n) which have solutions with nonvanishing a*b.

%C The solutions for odd numbers start with m = 15, 35, 39, 45, 51, 55, ... . See A329589 which is a proper subsequence of A257591. For example, m = 63 is not present in A329589.

%C There is a symmetry (a, b) <-> (m-a, m-b) == (-a, -b) (mod m) around the middle of the pairs in each row. Such pairs correspond to z and -z (mod m). Because of this symmetry one considers first a > b (a = b cannot occur for m >= 2, see above), and later adds the a < b solutions.

%C Obvious solutions for each solvable even m are for m/2 odd (m/2+1, m/2) with companion (m/2-1, m/2), and in addition, if m/(2^e2) is odd, for e2 >= 2, (m-1, m/2) with companion (1, m/2) (for m = 4 this coincides with the first case).

%C For (positive) even m, m = 2*M, there is the following connection to Pythagorean triples (X, Y, Z). We say triples, not triangles, because X may also become negative. Eq.(i) from above becomes (i') a^2 - (b^2 + 1) = qhat*2*M, with integer qhat. This implies that a and b have opposite parity, i.e., (-1)^(a+b) = -1. The other eq. (ii) is now (ii') a*b = q*M, with integer q. Thus (q, qhat) = (Y/m, (X-1)/m).

%C Case A) Primitive Pythagorean triples (pPT). If a > b and gcd(a, b) = 1 then the conditions for primitive Pythagorean triples are satisfied. Set 0 < X = a^2 - b^2 = 1 + 2*qhat*M, 0 < Y = 2*a*b = 2*q*M, 0 < Z = a^2 + b^2 = r^2 (Y is even, X is odd, and Z is odd). One could interchange the role of X and Y (for triangles a catheti exchange). Note that the radius r is in general not an integer. E.g., for m=4 (a, b) = (3, 2) has r = sqrt(13), (q, qhat) = (3, 1), pPT = (5, 12, 13).

%C Note that the companion triple for a < b (z -> -z) has negative X. In this example the companion of (a, b) = (3, 2) is (-a, -b) (mod 4) == (4-3, 4-2) = (1, 2), and the companion pPT = (-3, 4, 5).

%C Case B) m even, a > b, (-1)^(a+b) = -1, gcd(a, b) = g >= 2, leads to imprimitive Pythagorean triples (ipPT) for solutions of (i') and (ii'). The first example appears for m = 20, M = 10, (a, b) = (15, 12), g = 3, (q, qhat) = (18, 4), r = 3*sqrt(41), ipPT = (81, 360, 369) = (3^2)*(9, 40, 41). The companion has (a, b) = (5, 8), which is primitive, (q, qhat) = (4, -2), with pPT = (-39, 80, 89).

%C In A226746 the m values with more than two representative solutions of z^2 == +1 (mod m) are given. For the corresponding solutions one has also to consider the irregular triangle A329588(n).

%C The number of all representative solutions z^2 == +1 (mod m), for m >= 1, is found by combining A329586 and A329588, and is given in A227091.

%F Row n, with m = m(n), of this irregular triangle T(n, k), with row length A329588(n), lists all pairs (a, b) which solve z^2 == +1 (mod m), with z = a + b*i, and nonvanishing a*b, sorted with a > b pairs in both halves in the example separated by a | symbol) first with increasing a, then increasing b.

%e The irregular triangle T(n, k) begins: (A | symbol separates a > b and a < b pairs, a star indicates that a pair is not relatively prime. For n = 10, 12 and 15 two rows are given with corresponding q >= 7.)

%e n,   m \ q     1            2         3       4        5        6  ...

%e -----------------------------------------------------------------------

%e 1,   4:     (3,2)   |    (1,2)

%e 2,   6:     (4,3)   |    (2,3)

%e 3,   8:     (5,4)        (7,4) |   (1,4)   (3,4)

%e 4,  10:     (5,2)        (6,5) |   (4,5)   (5,8)

%e 5,  12:     (7,6)       (11,6) |   (1,6)   (5,6)

%e 6,  14:     (8,7)   |    (6,7)

%e 7,  15:     (5,3)       (10,3) |  (5,12) (10,12)*

%e 8,  16:     (9,8)       (15,8) |   (1,8)   (7,8)

%e 9,  18:    (10,9)   |    (8,9)

%e 10, 20:     (5,2)      (11,10)    (15,2)  (15,8)  (15,12)* (19,10) |

%e            (1,10)        (5,8)    (5,12)  (5,18)   (9,10)  (15,18)*

%e 11, 22:   (12,11)   |  (10,11)

%e 12, 24:   (13,12)      (17,12)   (19,12) (23,12) |

%e            (1,12)       (5,12)    (7,12) (11,12)

%e 13, 26:   (13,18)      (14,13) | (12,13)  (13,8)

%e 14, 28:   (15,14)      (27,14) |  (1,14) (13,14)

%e 15, 30:    (10,3)      (16,15)    (20,3) (25,12)  (25,18)  (26,15)

%e            (4,15)       (5,12)    (5,18) (10,27)  (14,15)  (20,27)

%e 16, 32:   (17,16)      (31,16) |  (1,16) (15,16)

%e 17, 34:    (17,4)      (18,17) | (16,17) (17,30)

%e 18, 35:    (15,7)       (20,7) | (15,28) (20,28)

%e ...

%e ----------------------------------------------------------------------------

%e n=1, m=4: (1 + 2*i)^2 = (1 - 4) + 2*2*i == -3 (mod 4) == 1 (mod 4).

%e           (3 + 2*i)^2 = (9 - 4) + 12*i == 1 (mod 4).

%e ----------------------------------------------------------------------------

%e For even m the Pythagorean triples (X,Y,Z) are:

%e m\    pPT and  ipPT*, also with companions with negative X, separated by a |

%e ---------------------------------------------------------------------------

%e 4:         (5,12,13)  |     (-3, 4, 5)

%e 6:         (7,24,25)  |     (-5,12,13)

%e 8:         (9,40,41)        (33,56,65)   |       (-15,8,17)       (-7,24,25)

%e 10:       (21,20,29)        (11,60,61)   |       (-9,40,41)      (-39,80,89)

%e 12:       (13,84,85)      (85,132,157)   |      (-35,12,37)      (-11,60,61)

%e 14:     (15,112,113)  |    (-13,84,85)

%e 16:     (17,144,145)     (161,240,289)   |      (-63,16,65)    (-15,112,113)

%e 18:     (19,180,181)  |  (-17,144,145)

%e 20:     (21, 20, 29)      (21,220,221)         (221,60,229)

%e        (161,240,289)     (81, 360,369)*       (261,380,461) |

%e         (-99,20,101)       (-39,80,89)       (-119,120,169)

%e       (-299,180,349)     (-19,180,181)        (-99,540,549)*

%e 22:     (23,264,265)  |  (-21,220,221)

%e 24:     (25,312,313)     (145,408,433)        (217,456,505)    (385,552,673)

%e        (-143,24,145)    (-119,120,169)        (-95,168,193)    (-23,264,265)

%e 26:    (105,208,233)      (27,364,365)   |    (-25,312,313)   (-155,468,493)

%e 28:     (29,420,421)     (533,756,925)   |    (-195,28,197)    (-27,364,365)

%e 30:      (91,60,109)      (31,480,481)        (391,120,409)

%e        (481,600,769)     (301,900,949)        (451,780,901) |

%e       (-209,120,241)    (-119,120,169)       (-299,180,349)

%e       (-629,540,829)     (-29,420,421)     (-329,1080,1129)

%e 32:     (33,544,545)    (705,992,1217)   |    (-255,32,257)    (-31,480,481)

%e 34:    (273,136,305)      (35,612,613)   |    (-33,544,545) (-611,1020,1189)

%e ...

%e ----------------------------------------------------------------------------

%Y Cf. A226746, A329585, A329586, A329588, A329589, A227091.

%K nonn,tabf

%O 1,1

%A _Wolfdieter Lang_, Dec 14 2019