================================================================ WHAT: ---------------------------------------------------------------- links: http://en.wikipedia.org/wiki/Cyclic_number http://en.wikipedia.org/wiki/Transposable_integer http://en.wikipedia.org/wiki/Repeating_decimal http://en.wikipedia.org/wiki/Cyclic_permutation_of_integer http://en.wikipedia.org/wiki/Parasitic_number http://mathworld.wolfram.com/CyclicNumber.html http://en.wikipedia.org/wiki/Midy%27s_Theorem http://mathworld.wolfram.com/MidysTheorem.html ---------------------------------------------------------------- 0. Generalization of 0.142857 for 1/7, 2/7, ... 6/7 in base 10. These are cyclic permutations of one another. Also note that 1,2,4,5,7,8 are units mod 9 (=b-1). Notation: n=7 b=10 k=1,2,3,4,5,6. Note that cyclic numbers (wikipedia xref) take n prime. I don't require that. "A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857." "The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are: * 142857 × 3 = 428571 (shifts cyclically one place left) * 142857 × 5 = 714285 (shifts cyclically one place right) * 128205 × 4 = 512820 (shifts cyclically one place right) * 076923 × 9 = 692307 (shifts cyclically two places left) These specific integers, known as transposable integers, can be but are not always cyclic numbers." --> this is info. ---------------------------------------------------------------- 1a. When do k/n's, for (k,n)=1, have: * period (in base-b expansion) equal to phi(n), and * all values cyclic permutations of one another? --> full period iff (?) b is primitive mod n (which in turn requires that n's unit group be cyclic -- cf. n=21). 1b. When that does occur, what properties do those phi(n) digits (the "digit set" of n and b, d(n,b)) have? --> none that i can tell ... 142857 being the unit group of 9 appears to be a coincidence ... :/ ---------------------------------------------------------------- 2. For shorter periods, how do the digit-sets partition (e.g. in relation to n and b), and what is the mapping of k's to partitions? --> mapping to partitions are cosets of mod n (orbit and co-orbits). ---------------------------------------------------------------- 3a. Fixing n, varying b, what periods can exist? --> They are all divisors of phi(n). This is because the period, f(n,b), is the smallest positive integer e such that n divides (b^e-1), i.e. b^e-1 == 0 mod n, i.e. b^e == 1 mod n: so, the group-theoretic period of b mod n. 3b. Fixing b, varying n, what periods can exist? In particular, why do there seem to be no periods of 8 in base 16? x-> this needs work. ---------------------------------------------------------------- 4. Using b=16, can I find n so that period is 8 and digits are relatively prime to 15 (namely, 1 2 4 7 8 b d e)? (This is analog of n=7,b=10.) If not, why not? --> no, since from 1a. b is square and non-trivial unit groups (needs xref ...) have even order. ---------------------------------------------------------------- 5. What's the relationship between add order (which cuts across digit-sets) and shift order (which preserves digit sets)? E.g. for n=7,b=10, shift order is the same as multiplication by 3 (i.e. tripled add order). --> Left shift is, visibly, multiplication by b, mod b^p-1. Multiplication by b is the same as by b mod n. ---------------------------------------------------------------- 6. Note for shift order: let m1 = (b^p-1)/n (i.e. digits of 1/n in base b). Let t(m1) = trailing digit of m1. Then the first shift is m2 = (m1 + t(m1)*(b^p-1)) / b but m1 = (b^p-1)/n so m2 = (m1 + t(m1)*n*m1) / b m2 = (1 + t(m1)*n) * m1/b so t(m1)*n == -1 (mod b). ---------------------------------------------------------------- 7. Half-period shifts: 142+857=999. How does this generalize (if at all)? --> Midy's theorem. Does it apply for b != 10? Look for counterexamples. ================================================================ (1:4) n=5 base=7: 0.1254 (1:4) n=5 base=8: 0.1463 (1:10) n=11 base=8: 0.0564272135 (2:3) n=7 base=11: 0.163; 0.479 (1:4) n=5 base=12: 0.2497 (1:6) n=7 base=12: 0.186a35 (13568a missing 2479b) (1:4) n=5 base=13: 0.27a5 (1:12) n=11 base=13: 0.12495ba837 (missing 6) (1:6) n=9 base=14: 0.17ac63 (1367ac missing 24589b) (1:12) n=13 base=15: 0.124936dca5b8 (missing 7) (2:3) n=7 base=16: 0.249; 0.6db (2:3) n=9 base=16: 0.1c7; 0.38e (2:5) n=11 base=16: 0.1745d; 0.2e8ba (4:3) n=13 base=16: 0.13b; 0.276; 0.4ec; 0.89d (2:9) n=19 base=16: 0.0d79435e5; 0.1af286bca ---------------------------------------------------------------- totients: 7 8 9 10 11 12 13 14 15 16 17 18 19 ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- 6 4 6 4 10 4 12 6 8 8 16 6 18 row labels: n column labels: base cells: period of n in base b n\b 7 8 9 10 11 12 13 14 15 16 --- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 2 1 1 1 1 1 3 1 2 1 2 1 2 1 4 2 1 2 1 2 5 4 4 2 1 4 4 2 1 6 1 2 1 7 1 3 6 3 6 2 1 3 8 2 1 2 2 2 9 3 2 1 6 3 6 3 10 4 2 1 4 11 10 10 5 2 1 10 5 5 5 12 2 2 1 13 12 4 3 6 12 2 1 12 3 14 3 3 2 1 15 4 4 2 4 2 1 16 2 2 4 4 2 17 16 8 8 16 16 16 4 16 8 2 18 3 6 3 19 3 6 9 18 3 6 18 18 18 9 20 4 2 2 4 21 2 6 6 2 3 22 10 5 10 5 23 22 11 11 22 22 11 11 22 22 11 24 2 2 2 25 4 20 10 5 20 20 10 5 26 12 3 12 12 27 9 6 3 18 9 18 9 28 3 6 2 2 29 7 28 14 28 28 4 14 28 28 7 30 4 2 4 31 15 5 15 15 30 30 30 15 10 5 32 4 4 8 8 2 33 10 10 2 10 10 5 34 16 8 16 4 8 35 4 6 3 12 4 3 36 6 6 3 37 9 12 9 3 6 9 36 12 36 9 38 3 9 3 18 18 39 12 4 6 12 2 3 40 4 2 2 4 41 40 20 4 5 40 40 40 8 40 5 42 6 2 43 6 14 21 21 7 42 21 21 21 7 44 10 5 10 10 45 12 4 6 12 6 3 46 22 11 22 11 22 47 23 23 23 46 46 23 46 23 46 23 48 2 4 4 49 7 21 42 21 42 14 7 21 50 4 10 5 20 51 16 8 16 16 4 16 2 52 12 3 12 12 53 26 52 26 13 26 52 13 52 13 13 54 9 18 9 55 20 20 10 4 20 10 5 56 3 6 2 2 57 3 6 18 6 18 18 9 58 7 14 28 14 28 59 29 58 29 58 58 29 58 58 29 29 60 4 2 4 ================================================================ ================================================================ n=7 b=10 p=6 f=6 r=1 1/7 = .142857 1**2 = 1 qr=1 3/7 = .428571 3**2 = 2 qr=0 2/7 = .285714 2**2 = 4 qr=1 6/7 = .857142 6**2 = 1 qr=0 4/7 = .571428 4**2 = 2 qr=1 5/7 = .714285 5**2 = 4 qr=0 n=13 b=10 p=6 f=12 r=2 1/13 = .076923 1**2 = 1 qr=1 10/13 = .769230 10**2 = 9 qr=1 9/13 = .692307 9**2 = 3 qr=1 12/13 = .923076 12**2 = 1 qr=1 3/13 = .230769 3**2 = 9 qr=1 4/13 = .307692 4**2 = 3 qr=1 2/13 = .153846 2**2 = 4 qr=0 7/13 = .538461 7**2 = 10 qr=0 5/13 = .384615 5**2 = 12 qr=0 11/13 = .846153 11**2 = 4 qr=0 6/13 = .461538 6**2 = 10 qr=0 8/13 = .615384 8**2 = 12 qr=0 1/13 = .076923 1**2 = 1 qr=1 B0 2/13 = .153846 2**2 = 4 qr=0 A0 3/13 = .230769 3**2 = 9 qr=1 B4 4/13 = .307692 4**2 = 3 qr=1 B5 5/13 = .384615 5**2 = 12 qr=0 A2 6/13 = .461538 6**2 = 10 qr=0 A4 7/13 = .538461 7**2 = 10 qr=0 A1 8/13 = .615384 8**2 = 12 qr=0 A5 9/13 = .692307 9**2 = 3 qr=1 B2 10/13 = .769230 10**2 = 9 qr=1 B1 11/13 = .846153 11**2 = 4 qr=0 A3 12/13 = .923076 12**2 = 1 qr=1 B3 n=17 b=10 p=16 f=16 r=1 1/17 = .0588235294117647 1**2 = 1 qr=1 10/17 = .5882352941176470 10**2 = 15 qr=0 15/17 = .8823529411764705 15**2 = 4 qr=1 14/17 = .8235294117647058 14**2 = 9 qr=0 4/17 = .2352941176470588 4**2 = 16 qr=1 6/17 = .3529411764705882 6**2 = 2 qr=0 9/17 = .5294117647058823 9**2 = 13 qr=1 5/17 = .2941176470588235 5**2 = 8 qr=0 16/17 = .9411764705882352 16**2 = 1 qr=1 7/17 = .4117647058823529 7**2 = 15 qr=0 2/17 = .1176470588235294 2**2 = 4 qr=1 3/17 = .1764705882352941 3**2 = 9 qr=0 13/17 = .7647058823529411 13**2 = 16 qr=1 11/17 = .6470588235294117 11**2 = 2 qr=0 8/17 = .4705882352941176 8**2 = 13 qr=1 12/17 = .7058823529411764 12**2 = 8 qr=0 --> orbit of 10 n=19 b=10 p=18 f=18 r=1 1/19 = .052631578947368421 1**2 = 1 qr=1 10/19 = .526315789473684210 10**2 = 5 qr=0 5/19 = .263157894736842105 5**2 = 6 qr=1 12/19 = .631578947368421052 12**2 = 11 qr=0 6/19 = .315789473684210526 6**2 = 17 qr=1 3/19 = .157894736842105263 3**2 = 9 qr=0 11/19 = .578947368421052631 11**2 = 7 qr=1 15/19 = .789473684210526315 15**2 = 16 qr=0 17/19 = .894736842105263157 17**2 = 4 qr=1 18/19 = .947368421052631578 18**2 = 1 qr=0 9/19 = .473684210526315789 9**2 = 5 qr=1 14/19 = .736842105263157894 14**2 = 6 qr=0 7/19 = .368421052631578947 7**2 = 11 qr=1 13/19 = .684210526315789473 13**2 = 17 qr=0 16/19 = .842105263157894736 16**2 = 9 qr=1 8/19 = .421052631578947368 8**2 = 7 qr=0 4/19 = .210526315789473684 4**2 = 16 qr=1 2/19 = .105263157894736842 2**2 = 4 qr=0 n=21 b=10 p=6 f=12 r=2 1/21 = .047619 1**2 = 1 qr=1 10/21 = .476190 10**2 = 16 qr=0 16/21 = .761904 16**2 = 4 qr=1 13/21 = .619047 13**2 = 1 qr=0 4/21 = .190476 4**2 = 16 qr=1 19/21 = .904761 19**2 = 4 qr=0 2/21 = .095238 2**2 = 4 qr=0 20/21 = .952380 20**2 = 1 qr=0 11/21 = .523809 11**2 = 16 qr=0 5/21 = .238095 5**2 = 4 qr=0 8/21 = .380952 8**2 = 1 qr=0 17/21 = .809523 17**2 = 16 qr=0 n=23 b=10 p=22 f=22 r=1 1/23 = .0434782608695652173913 1**2 = 1 qr=1 10/23 = .4347826086956521739130 10**2 = 8 qr=0 8/23 = .3478260869565217391304 8**2 = 18 qr=1 11/23 = .4782608695652173913043 11**2 = 6 qr=0 18/23 = .7826086956521739130434 18**2 = 2 qr=1 19/23 = .8260869565217391304347 19**2 = 16 qr=0 6/23 = .2608695652173913043478 6**2 = 13 qr=1 14/23 = .6086956521739130434782 14**2 = 12 qr=0 2/23 = .0869565217391304347826 2**2 = 4 qr=1 20/23 = .8695652173913043478260 20**2 = 9 qr=0 16/23 = .6956521739130434782608 16**2 = 3 qr=1 22/23 = .9565217391304347826086 22**2 = 1 qr=0 13/23 = .5652173913043478260869 13**2 = 8 qr=1 15/23 = .6521739130434782608695 15**2 = 18 qr=0 12/23 = .5217391304347826086956 12**2 = 6 qr=1 5/23 = .2173913043478260869565 5**2 = 2 qr=0 4/23 = .1739130434782608695652 4**2 = 16 qr=1 17/23 = .7391304347826086956521 17**2 = 13 qr=0 9/23 = .3913043478260869565217 9**2 = 12 qr=1 21/23 = .9130434782608695652173 21**2 = 4 qr=0 3/23 = .1304347826086956521739 3**2 = 9 qr=1 7/23 = .3043478260869565217391 7**2 = 3 qr=0 n=27 b=10 p=3 f=18 r=6 1/27 = .037 1**2 = 1 qr=1 10/27 = .370 10**2 = 19 qr=1 19/27 = .703 19**2 = 10 qr=1 2/27 = .074 2**2 = 4 qr=0 20/27 = .740 20**2 = 22 qr=0 11/27 = .407 11**2 = 13 qr=0 4/27 = .148 4**2 = 16 qr=1 13/27 = .481 13**2 = 7 qr=1 22/27 = .814 22**2 = 25 qr=1 5/27 = .185 5**2 = 25 qr=0 23/27 = .851 23**2 = 16 qr=0 14/27 = .518 14**2 = 7 qr=0 7/27 = .259 7**2 = 22 qr=1 16/27 = .592 16**2 = 13 qr=1 25/27 = .925 25**2 = 4 qr=1 8/27 = .296 8**2 = 10 qr=0 26/27 = .962 26**2 = 1 qr=0 17/27 = .629 17**2 = 19 qr=0 n=29 b=10 p=28 f=28 r=1 1/29 = .0344827586206896551724137931 1**2 = 1 qr=1 10/29 = .3448275862068965517241379310 10**2 = 13 qr=0 13/29 = .4482758620689655172413793103 13**2 = 24 qr=1 14/29 = .4827586206896551724137931034 14**2 = 22 qr=0 24/29 = .8275862068965517241379310344 24**2 = 25 qr=1 8/29 = .2758620689655172413793103448 8**2 = 6 qr=0 22/29 = .7586206896551724137931034482 22**2 = 20 qr=1 17/29 = .5862068965517241379310344827 17**2 = 28 qr=0 25/29 = .8620689655172413793103448275 25**2 = 16 qr=1 18/29 = .6206896551724137931034482758 18**2 = 5 qr=0 6/29 = .2068965517241379310344827586 6**2 = 7 qr=1 2/29 = .0689655172413793103448275862 2**2 = 4 qr=0 20/29 = .6896551724137931034482758620 20**2 = 23 qr=1 26/29 = .8965517241379310344827586206 26**2 = 9 qr=0 28/29 = .9655172413793103448275862068 28**2 = 1 qr=1 19/29 = .6551724137931034482758620689 19**2 = 13 qr=0 16/29 = .5517241379310344827586206896 16**2 = 24 qr=1 15/29 = .5172413793103448275862068965 15**2 = 22 qr=0 5/29 = .1724137931034482758620689655 5**2 = 25 qr=1 21/29 = .7241379310344827586206896551 21**2 = 6 qr=0 7/29 = .2413793103448275862068965517 7**2 = 20 qr=1 12/29 = .4137931034482758620689655172 12**2 = 28 qr=0 4/29 = .1379310344827586206896551724 4**2 = 16 qr=1 11/29 = .3793103448275862068965517241 11**2 = 5 qr=0 23/29 = .7931034482758620689655172413 23**2 = 7 qr=1 27/29 = .9310344827586206896551724137 27**2 = 4 qr=0 9/29 = .3103448275862068965517241379 9**2 = 23 qr=1 3/29 = .1034482758620689655172413793 3**2 = 9 qr=0 ================================================================ n=7 b=16 p=3 f=6 r=2 1/7 = .249 1**2 = 1 qr=1 2/7 = .492 2**2 = 4 qr=1 4/7 = .924 4**2 = 2 qr=1 3/7 = .6db 3**2 = 2 qr=0 6/7 = .db6 6**2 = 1 qr=0 5/7 = .b6d 5**2 = 4 qr=0 n=9 b=16 p=3 f=6 r=2 1/9 = .1c7 1**2 = 1 qr=1 7/9 = .c71 7**2 = 4 qr=1 4/9 = .71c 4**2 = 7 qr=1 2/9 = .38e 2**2 = 4 qr=0 5/9 = .8e3 5**2 = 7 qr=0 8/9 = .e38 8**2 = 1 qr=0 n=11 b=16 p=5 f=10 r=2 1/11 = .1745d 1**2 = 1 qr=1 5/11 = .745d1 5**2 = 3 qr=1 3/11 = .45d17 3**2 = 9 qr=1 4/11 = .5d174 4**2 = 5 qr=1 9/11 = .d1745 9**2 = 4 qr=1 2/11 = .2e8ba 2**2 = 4 qr=0 10/11 = .e8ba2 10**2 = 1 qr=0 6/11 = .8ba2e 6**2 = 3 qr=0 8/11 = .ba2e8 8**2 = 9 qr=0 7/11 = .a2e8b 7**2 = 5 qr=0 n=13 b=16 p=3 f=12 r=4 1/13 = .13b 1**2 = 1 qr=1 3/13 = .3b1 3**2 = 9 qr=1 9/13 = .b13 9**2 = 3 qr=1 2/13 = .276 2**2 = 4 qr=0 6/13 = .762 6**2 = 10 qr=0 5/13 = .627 5**2 = 12 qr=0 4/13 = .4ec 4**2 = 3 qr=1 12/13 = .ec4 12**2 = 1 qr=1 10/13 = .c4e 10**2 = 9 qr=1 7/13 = .89d 7**2 = 10 qr=0 8/13 = .9d8 8**2 = 12 qr=0 11/13 = .d89 11**2 = 4 qr=0 n=19 b=16 p=9 f=18 r=2 1/19 = .0d79435e5 1**2 = 1 qr=1 16/19 = .d79435e50 16**2 = 9 qr=1 9/19 = .79435e50d 9**2 = 5 qr=1 11/19 = .9435e50d7 11**2 = 7 qr=1 5/19 = .435e50d79 5**2 = 6 qr=1 4/19 = .35e50d794 4**2 = 16 qr=1 7/19 = .5e50d7943 7**2 = 11 qr=1 17/19 = .e50d79435 17**2 = 4 qr=1 6/19 = .50d79435e 6**2 = 17 qr=1 2/19 = .1af286bca 2**2 = 4 qr=0 13/19 = .af286bca1 13**2 = 17 qr=0 18/19 = .f286bca1a 18**2 = 1 qr=0 3/19 = .286bca1af 3**2 = 9 qr=0 10/19 = .86bca1af2 10**2 = 5 qr=0 8/19 = .6bca1af28 8**2 = 7 qr=0 14/19 = .bca1af286 14**2 = 6 qr=0 15/19 = .ca1af286b 15**2 = 16 qr=0 12/19 = .a1af286bc 12**2 = 11 qr=0 n=21 b=16 p=3 f=12 r=4 1/21 = .0c3 1**2 = 1 qr=1 16/21 = .c30 16**2 = 4 qr=1 4/21 = .30c 4**2 = 16 qr=1 2/21 = .186 2**2 = 4 qr=0 11/21 = .861 11**2 = 16 qr=0 8/21 = .618 8**2 = 1 qr=0 5/21 = .3cf 5**2 = 4 qr=0 17/21 = .cf3 17**2 = 16 qr=0 20/21 = .f3c 20**2 = 1 qr=0 10/21 = .79e 10**2 = 16 qr=0 13/21 = .9e7 13**2 = 1 qr=0 19/21 = .e79 19**2 = 4 qr=0 n=23 b=16 p=11 f=22 r=2 1/23 = .0b21642c859 1**2 = 1 qr=1 16/23 = .b21642c8590 16**2 = 3 qr=1 3/23 = .21642c8590b 3**2 = 9 qr=1 2/23 = .1642c8590b2 2**2 = 4 qr=1 9/23 = .642c8590b21 9**2 = 12 qr=1 6/23 = .42c8590b216 6**2 = 13 qr=1 4/23 = .2c8590b2164 4**2 = 16 qr=1 18/23 = .c8590b21642 18**2 = 2 qr=1 12/23 = .8590b21642c 12**2 = 6 qr=1 8/23 = .590b21642c8 8**2 = 18 qr=1 13/23 = .90b21642c85 13**2 = 8 qr=1 5/23 = .37a6f4de9bd 5**2 = 2 qr=0 11/23 = .7a6f4de9bd3 11**2 = 6 qr=0 15/23 = .a6f4de9bd37 15**2 = 18 qr=0 10/23 = .6f4de9bd37a 10**2 = 8 qr=0 22/23 = .f4de9bd37a6 22**2 = 1 qr=0 7/23 = .4de9bd37a6f 7**2 = 3 qr=0 20/23 = .de9bd37a6f4 20**2 = 9 qr=0 21/23 = .e9bd37a6f4d 21**2 = 4 qr=0 14/23 = .9bd37a6f4de 14**2 = 12 qr=0 17/23 = .bd37a6f4de9 17**2 = 13 qr=0 19/23 = .d37a6f4de9b 19**2 = 16 qr=0 n=25 b=16 p=5 f=20 r=4 1/25 = .0a3d7 1**2 = 1 qr=1 16/25 = .a3d70 16**2 = 6 qr=1 6/25 = .3d70a 6**2 = 11 qr=1 21/25 = .d70a3 21**2 = 16 qr=1 11/25 = .70a3d 11**2 = 21 qr=1 2/25 = .147ae 2**2 = 4 qr=0 7/25 = .47ae1 7**2 = 24 qr=0 12/25 = .7ae14 12**2 = 19 qr=0 17/25 = .ae147 17**2 = 14 qr=0 22/25 = .e147a 22**2 = 9 qr=0 3/25 = .1eb85 3**2 = 9 qr=0 23/25 = .eb851 23**2 = 4 qr=0 18/25 = .b851e 18**2 = 24 qr=0 13/25 = .851eb 13**2 = 19 qr=0 8/25 = .51eb8 8**2 = 14 qr=0 4/25 = .28f5c 4**2 = 16 qr=1 14/25 = .8f5c2 14**2 = 21 qr=1 24/25 = .f5c28 24**2 = 1 qr=1 9/25 = .5c28f 9**2 = 6 qr=1 19/25 = .c28f5 19**2 = 11 qr=1 n=27 b=16 p=9 f=18 r=2 1/27 = .097b425ed 1**2 = 1 qr=1 16/27 = .97b425ed0 16**2 = 13 qr=1 13/27 = .7b425ed09 13**2 = 7 qr=1 19/27 = .b425ed097 19**2 = 10 qr=1 7/27 = .425ed097b 7**2 = 22 qr=1 4/27 = .25ed097b4 4**2 = 16 qr=1 10/27 = .5ed097b42 10**2 = 19 qr=1 25/27 = .ed097b425 25**2 = 4 qr=1 22/27 = .d097b425e 22**2 = 25 qr=1 2/27 = .12f684bda 2**2 = 4 qr=0 5/27 = .2f684bda1 5**2 = 25 qr=0 26/27 = .f684bda12 26**2 = 1 qr=0 11/27 = .684bda12f 11**2 = 13 qr=0 14/27 = .84bda12f6 14**2 = 7 qr=0 8/27 = .4bda12f68 8**2 = 10 qr=0 20/27 = .bda12f684 20**2 = 22 qr=0 23/27 = .da12f684b 23**2 = 16 qr=0 17/27 = .a12f684bd 17**2 = 19 qr=0 n=29 b=16 p=7 f=28 r=4 1/29 = .08d3dcb 1**2 = 1 qr=1 16/29 = .8d3dcb0 16**2 = 24 qr=1 24/29 = .d3dcb08 24**2 = 25 qr=1 7/29 = .3dcb08d 7**2 = 20 qr=1 25/29 = .dcb08d3 25**2 = 16 qr=1 23/29 = .cb08d3d 23**2 = 7 qr=1 20/29 = .b08d3dc 20**2 = 23 qr=1 2/29 = .11a7b96 2**2 = 4 qr=0 3/29 = .1a7b961 3**2 = 9 qr=0 19/29 = .a7b9611 19**2 = 13 qr=0 14/29 = .7b9611a 14**2 = 22 qr=0 21/29 = .b9611a7 21**2 = 6 qr=0 17/29 = .9611a7b 17**2 = 28 qr=0 11/29 = .611a7b9 11**2 = 5 qr=0 4/29 = .234f72c 4**2 = 16 qr=1 6/29 = .34f72c2 6**2 = 7 qr=1 9/29 = .4f72c23 9**2 = 23 qr=1 28/29 = .f72c234 28**2 = 1 qr=1 13/29 = .72c234f 13**2 = 24 qr=1 5/29 = .2c234f7 5**2 = 25 qr=1 22/29 = .c234f72 22**2 = 20 qr=1 8/29 = .469ee58 8**2 = 6 qr=0 12/29 = .69ee584 12**2 = 28 qr=0 18/29 = .9ee5846 18**2 = 5 qr=0 27/29 = .ee58469 27**2 = 4 qr=0 26/29 = .e58469e 26**2 = 9 qr=0 10/29 = .58469ee 10**2 = 13 qr=0 15/29 = .8469ee5 15**2 = 22 qr=0