A329587 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.

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, 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, 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

Offset: 1

Author

Wolfdieter Lang, Dec 14 2019

Keywords

• nonn
• tabf

Comments

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

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).

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.

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.

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

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.

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.

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).

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).

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).

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).

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).

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).

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.

Examples

			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.)
n,   m \ q     1            2         3       4        5        6  ...
-----------------------------------------------------------------------
1,   4:     (3,2)   |    (1,2)
2,   6:     (4,3)   |    (2,3)
3,   8:     (5,4)        (7,4) |   (1,4)   (3,4)
4,  10:     (5,2)        (6,5) |   (4,5)   (5,8)
5,  12:     (7,6)       (11,6) |   (1,6)   (5,6)
6,  14:     (8,7)   |    (6,7)
7,  15:     (5,3)       (10,3) |  (5,12) (10,12)*
8,  16:     (9,8)       (15,8) |   (1,8)   (7,8)
9,  18:    (10,9)   |    (8,9)
10, 20:     (5,2)      (11,10)    (15,2)  (15,8)  (15,12)* (19,10) |
           (1,10)        (5,8)    (5,12)  (5,18)   (9,10)  (15,18)*
11, 22:   (12,11)   |  (10,11)
12, 24:   (13,12)      (17,12)   (19,12) (23,12) |
           (1,12)       (5,12)    (7,12) (11,12)
13, 26:   (13,18)      (14,13) | (12,13)  (13,8)
14, 28:   (15,14)      (27,14) |  (1,14) (13,14)
15, 30:    (10,3)      (16,15)    (20,3) (25,12)  (25,18)  (26,15)
           (4,15)       (5,12)    (5,18) (10,27)  (14,15)  (20,27)
16, 32:   (17,16)      (31,16) |  (1,16) (15,16)
17, 34:    (17,4)      (18,17) | (16,17) (17,30)
18, 35:    (15,7)       (20,7) | (15,28) (20,28)
...
----------------------------------------------------------------------------
n=1, m=4: (1 + 2*i)^2 = (1 - 4) + 2*2*i == -3 (mod 4) == 1 (mod 4).
          (3 + 2*i)^2 = (9 - 4) + 12*i == 1 (mod 4).
----------------------------------------------------------------------------
For even m the Pythagorean triples (X,Y,Z) are:
m\    pPT and  ipPT*, also with companions with negative X, separated by a |
---------------------------------------------------------------------------
4:         (5,12,13)  |     (-3, 4, 5)
6:         (7,24,25)  |     (-5,12,13)
8:         (9,40,41)        (33,56,65)   |       (-15,8,17)       (-7,24,25)
10:       (21,20,29)        (11,60,61)   |       (-9,40,41)      (-39,80,89)
12:       (13,84,85)      (85,132,157)   |      (-35,12,37)      (-11,60,61)
14:     (15,112,113)  |    (-13,84,85)
16:     (17,144,145)     (161,240,289)   |      (-63,16,65)    (-15,112,113)
18:     (19,180,181)  |  (-17,144,145)
20:     (21, 20, 29)      (21,220,221)         (221,60,229)
       (161,240,289)     (81, 360,369)*       (261,380,461) |
        (-99,20,101)       (-39,80,89)       (-119,120,169)
      (-299,180,349)     (-19,180,181)        (-99,540,549)*
22:     (23,264,265)  |  (-21,220,221)
24:     (25,312,313)     (145,408,433)        (217,456,505)    (385,552,673)
       (-143,24,145)    (-119,120,169)        (-95,168,193)    (-23,264,265)
26:    (105,208,233)      (27,364,365)   |    (-25,312,313)   (-155,468,493)
28:     (29,420,421)     (533,756,925)   |    (-195,28,197)    (-27,364,365)
30:      (91,60,109)      (31,480,481)        (391,120,409)
       (481,600,769)     (301,900,949)        (451,780,901) |
      (-209,120,241)    (-119,120,169)       (-299,180,349)
      (-629,540,829)     (-29,420,421)     (-329,1080,1129)
32:     (33,544,545)    (705,992,1217)   |    (-255,32,257)    (-31,480,481)
34:    (273,136,305)      (35,612,613)   |    (-33,544,545) (-611,1020,1189)
...
----------------------------------------------------------------------------
		

Crossrefs

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

Formula

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.