20220621, 03:34  #1 
Jun 2022
2×5 Posts 
Phi mod 11 is congruent 4 and 8
What about other irrational numbers. e seens to be 6.....what about pi.....Would the square root of 2 be an imaginary number?

20220621, 11:39  #2  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3·29·41 Posts 
Quote:
Phi (mod n) only exist when 5 is quadratic residue mod n (since Phi has sqrt(5)) and n is coprime to 2 (since Phi has 1/2) Code:
n Phi (mod n) 1 0 5 3 11 4 or 8 19 5 or 14 29 6 or 23 31 12 or 19 41 7 or 34 55 8 or 48 59 25 or 34 61 17 or 44 71 9 or 62 79 29 or 50 89 10 or 79 95 33 or 43 101 23 or 78 109 11 or 98 121 37 or 85 131 12 or 119 139 63 or 76 145 23 or 93 149 40 or 109 151 28 or 123 155 43 or 143 179 74 or 105 181 14 or 169 Last fiddled with by sweety439 on 20220621 at 11:49 

20220621, 11:42  #3  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3·29·41 Posts 
Quote:
the square root of 2 is imaginary number if and only if 2 is not quadratic residue to your modulo, e.g. sqrt(2) mod 3, in Z3, you will find a field extension, i.e. Z3(sqrt(2)) 

20220621, 18:58  #4  
Feb 2017
Nowhere
2^{3}×11×67 Posts 
Quote:
x^2  x  1 == 0 (mod p) If p = 5 there is one repeated root. For p == 2 or 3 (mod 5) there are no roots. However, at least for the prime moduli other than 5 and 11, one of the residues in the quoted portion above appears to be off by 1. Code:
? forprime(p=2,200,r=p%5;if(r==1r==4,M=factormod(x^2x1,p);v=subst(M[,1],x,0)~;print(p" "lift(v)))) 11 [8, 4] 19 [15, 5] 29 [24, 6] 31 [19, 13] 41 [35, 7] 59 [34, 26] 61 [44, 18] 71 [63, 9] 79 [50, 30] 89 [80, 10] 101 [79, 23] 109 [99, 11] 131 [120, 12] 139 [76, 64] 149 [109, 41] 151 [124, 28] 179 [105, 75] 181 [168, 14] 191 [103, 89] 199 [138, 62] Last fiddled with by Dr Sardonicus on 20220621 at 19:17 

20220622, 03:04  #5  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3×29×41 Posts 
Quote:
Code:
1: 0, 5: 3, 11: 4, 8, 19: 5, 15, 29: 6, 24, 31: 13, 19, 41: 7, 35, 55: 8, 48, 59: 26, 34, 61: 18, 44, 71: 9, 63, 79: 30, 50, 89: 10, 80, 95: 43, 53, 101: 23, 79, 109: 11, 99, 121: 37, 85, 131: 12, 120, 139: 64, 76, 145: 53, 93, 149: 41, 109, 151: 28, 124, 155: 13, 143, 179: 75, 105, 181: 14, 168, 191: 89, 103, 199: 62, 138, 205: 48, 158, 209: 15, 81, 129, 195, 211: 33, 179, 229: 82, 148, 239: 16, 224, 241: 52, 190, 251: 118, 134, 269: 72, 198, 271: 17, 255, 281: 38, 244, 295: 93, 203, 305: 18, 288, 311: 59, 253, 319: 140, 151, 169, 180, 331: 117, 215, 341: 19, 81, 261, 323, 349: 144, 206, 355: 63, 293, 359: 106, 254, 361: 43, 319, 379: 20, 360, 389: 152, 238, 395: 188, 208, 401: 112, 290, 409: 130, 280, 419: 21, 399, 421: 111, 311, 431: 91, 341, 439: 70, 370, 445: 188, 258, 449: 166, 284, 451: 48, 158, 294, 404, 461: 22, 440, 479: 229, 251, 491: 74, 418, 499: 225, 275, 505: 23, 483, 509: 122, 388, 521: 100, 422, 541: 173, 369, 545: 208, 338, 551: 24, 53, 499, 528, 569: 233, 337, 571: 274, 298, 589: 81, 205, 385, 509, 599: 25, 575, 601: 137, 465, 605: 158, 448, 619: 243, 377, 631: 110, 522, 641: 279, 363, 649: 26, 85, 565, 624, 655: 143, 513, 659: 201, 459, 661: 58, 604, 671: 140, 323, 349, 532, 691: 222, 470, 695: 203, 493, 701: 27, 675, 709: 171, 539, 719: 330, 390, 739: 119, 621, 745: 258, 488, 751: 211, 541, 755: 28, 728, 761: 92, 670, 769: 339, 431, 779: 281, 376, 404, 499, 781: 63, 151, 631, 719, 809: 343, 467, 811: 29, 783, 821: 213, 609, 829: 96, 734, 839: 342, 498, 841: 227, 615, 859: 277, 583, 869: 30, 129, 741, 840, 881: 327, 555, 895: 433, 463, 899: 267, 354, 546, 633, 905: 168, 738, 911: 68, 844, 919: 317, 603, 929: 31, 899, 941: 228, 714, 955: 103, 853, 961: 199, 763, 971: 174, 798, 979: 169, 455, 525, 811, 991: 32, 960, 995: 138, 858, 

20220623, 15:59  #6  
Jun 2022
2·5 Posts 
Quote:


20220624, 05:40  #7 
Jun 2022
2×5 Posts 
so I guess it would need to be N^11/N!.....to avoid division by 11.....

20220624, 05:57  #8 
Sep 2002
Database er0rr
1000010011100_{2} Posts 
I think you meant 11^N/(N!).

20220624, 05:57  #9  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
110111101111_{2} Posts 
Quote:


20220624, 12:28  #10 
Jun 2022
2×5 Posts 
Yes thank you
11^0/0! + 11^1/1! + 11^2/2!....... I believe this is transendental but now that I think about it I think it will go to infinity......I am doubting that this converges on a number....sorry seems I am babbling now.... Last fiddled with by x13420x on 20220624 at 12:30 
20220624, 12:36  #11 
Jun 2022
A_{16} Posts 
Did you guys notice when you add the 2 results you get the modulo number plus 1? like 8 + 4 = 11 + 1

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