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2020-01-18, 11:50   #1
mart_r

Dec 2008
you know...around...

22×7×19 Posts
Patterns in primes that are primitive roots / Gaps in full-reptend primes

This started out as a search for gaps between cyclic or full-reptend primes, where 10 is a primitive root mod p (A001913):
Code:
gap  p
2  17
4  19
6  23
8  491
10  7
12  47
14  419
16  577
18  29
20  -
22  1789
...  ...
A gap of 20 seemed impossible to find. And indeed, gaps that are congruent to 20 (mod 40) are not admissible.
If 10 is not a square mod p and p-20 or p+20 is a prime, then 10 is a square mod p-20 or p+20 respectively.

In general, for primes p where r is a primitive root mod p, these gaps are all inadmissible:
Code:
 r  gaps congruent to
2  4 (mod 8)
3  4,6,8 (mod 12)
5  2,8 (mod 10)
6  8,12,16 (mod 24)
7  14 (mod 28)
8  2,4,8,10,12,14,16,20,22 (mod 24)
10  20 (mod 40)
11  22 (mod 44)
12  4,6,8 (mod 12)
13  none
14  28 (mod 56)
15  20,30,40 (mod 60)
17  none
18  4 (mod 8)
19  38 (mod 76)
20  2,8 (mod 10)
I have a vague idea that this can be checked via the Legendre or Jacobi symbol and that $(\frac{r}{p})=(-1)^{\frac{(r-1)(p-1)}{4}}(\frac{p}{r})$, but I'm kind of stuck here (especially when r is not prime). Would like to understand how to prove it, as it would also help to determine that 11 is the smallest number that can be a primitive root mod every prime of a prime quadruplet p+{0,2,6,8}, a result I stumbled upon several years ago.

Next, looking for record gaps between Wieferich primes...
Attached Files
 FRP gaps.txt (7.6 KB, 27 views)

Last fiddled with by mart_r on 2020-01-18 at 11:56

2020-01-18, 15:47   #2
Dr Sardonicus

Feb 2017
Nowhere

CB216 Posts

Quote:
 Originally Posted by mart_r I have a vague idea that this can be checked via the Legendre or Jacobi symbol and that $(\frac{r}{p})=(-1)^{\frac{(r-1)(p-1)}{4}}(\frac{p}{r})$, but I'm kind of stuck here (especially when r is not prime). Would like to understand how to prove it, as it would also help to determine that 11 is the smallest number that can be a primitive root mod every prime of a prime quadruplet p+{0,2,6,8}, a result I stumbled upon several years ago.
I assume r is a positive integer. Let r = d*f^2 where d is square free (in Pari-GP, d = core(r)). If d = 1, r is a perfect square. Bail out! Otherwise...

If d is not congruent to 1 (mod 4), d = 4*d.

Now, d is a "fundamental discriminant," and (apart from primes that may have divided r to even powers), is the least modulus for which the quadratic character (r/p) is equal to (p/d). Whether d is a quadratic residue (mod p) depends only on p (mod d).

Thus, the quadratic character of r (mod p) is (again, apart from primes p that divide r to even powers) equal to (p/d).

For example, if r = 10, we obtain d = 40.

You can compute the quadratic non-residues (mod d) [assuming d isn't very big] in Pari-GP as follows:

Code:
v=vector(eulerphi(d)/2);j=0;for(i=1,d-1,if(kronecker(i,d)==-1,j++;v[j]=i))
Since d is even, the values in v are all odd, so the differences between any two of them are even.

But things get a bit tricky. If the odd part of d is congruent to 1 (mod 4), then if k is one of the numbers in v, so is d - k. Thus, the differences v[j] - v[i] for j > i will give all possible gaps between quadratic non-residues (mod d).

However, if the odd part of d is congruent to 3 (mod 4) this may not be true. (It happens to be true for d = 4*3 = 12, where v = [5, 11] and the only difference is 6). I checked the cases d = 4*7 and 4*11, and in both cases the further differences d + v[i] - v[j] for j > i gave all the even gaps (mod d) that weren't given by v[j] - v[i] with j > i. For r = 11, d = 44, here are the results:

Code:
? d=44;v = vector(eulerphi(d)/2);j=0;for(i=1,d-1,if(kronecker(i,d)==-1,j++;v[j]=i))

? w=vector(#v*(#v-1));k=0;for(i=1,#v-1,for(j=i+1,#v,m=v[j]-v[i];k++;w[k]=m;m=d-m;k++;w[k]=m));w=vecsort(w);

? w
%3 = [2, 2, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 12, 12, 12, 12, 14, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 26, 26, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 32, 32, 32, 32, 34, 34, 34, 34, 36, 36, 36, 36, 38, 38, 38, 38, 40, 40, 40, 40, 42, 42, 42, 42]

I suspect the general result is known, but I am too lazy to track it down at the moment.

 2020-01-18, 22:16 #3 Dr Sardonicus     Feb 2017 Nowhere 62628 Posts Oh, boy, did I ever foul up! I did the wrong kronecker() calculation. I should have put the value of d first. So I started over, and cobbled together something that at least gives results. I'm sure it could be turned into something usable for at least very small values of interest. The following Pari-GP code does work, but it is a total kludge. I wouldn't use it on numbers d of any size. I didn't even bother with code to make the value of d "suitable," which I decided was values that were either 4 or 8 times odd square free numbers. Given a value r, d = 4*core(r). Instead, I stuck to code for a specific value of d while I worked out my other mistakes, and when I got something that actually worked, tried other values of d by recalling the previous command and filling in new values by hand. I found that for d = 4*3, the gaps 4, 6, and 8 did not occur as the differences of non-residues. For d = 4*5, the gaps 2, 8, 12, and 18 do not occur. d = 4*p, p = 13, 17, and 29, all gaps occur as differences of non-residues. For d = 4*p, p = 7, 11, 19, and 23, only the gap 2*p does not occur. For d = 8*3, the gaps 8, 12, and 16 do not occur. For d = 8*p, p = 5, 7, 11, 13, 17, 19, only the gap 4*p does not occur. Code: { d=120; v=vector(eulerphi(d)/2); j=0; for(i=1,d-1,if(kronecker(d,i)==-1,j++;v[j]=i)); w=vector(#v*(#v-1)); k=0; for(i=1,#v-1,for(j=i+1,#v,k++;w[k]=v[j]-v[i];k++;w[k]=d+v[i]-v[j])); gaplist=listsort(List(w),1); if(#gaplist==d/2-1, print(); print("For d = "d" all gaps occur"); return(), l=d/2-#gaplist-1; ng=vector(l); g=vector(d/2-1,i,2*i); for(i=1,#gaplist, r=gaplist[i]; g[r/2]=0 ); j=0; for(i=1,d/2-1, if(g[i]>0, j++; ng[j]=g[i]) ); ); print(); print("d = "d); print("Vector of gaps that are not differences of non-residues is "ng) } The output is d = 120 Vector of gaps that are not differences of non-residues is [40, 60, 80]
 2020-01-19, 14:28 #4 mart_r     Dec 2008 you know...around... 22·7·19 Posts Thanks, that helps a lot! Based on your first post, I wrote my own program, and the results agree with yours. At first I was a bit confused about the results for r=8, but since 8 is a perfect power it's a slightly different story where kronecker alone doesn't help. (All primitive roots mod 8 are either 3, 5, or 11 mod 24.) I can't say that I fully understand all the maths behind it yet, but I can relate to the following: A non-perfect power r can only be a primitive root mod p if it's not a square mod (p mod 4r). Thus, to check which gaps are inadmissible, we only have to check all the differences between the set of odd numbers 2n+1 < 8r for which r is a nonsquare mod 2n+1. That, in principle, answers my main questions about this subject. But I'm going to work my way through the proof of quadratic reciprocity once more. I'm reading "Elementary Number Theory" by W. Stein.
 2020-01-19, 14:43 #5 sweety439     Nov 2016 22·13·37 Posts For gaps between primes p which 2 is primitive root mod p: Code: 2,3 6,5 8,29 10,19 14,149 16,37 18,83 22,421 24,107 26,587 30,317 32,2099 34,619 38,2621 40,1693 42,227 46,1381 48,709 50,3203 54,2477 56,4547 58,12979 62,4157 64,4723 66,1307 70,4021 72,947 74,1787 78,5573 80,12659 82,23251 86,20357 88,9949 90,13523 94,18493 96,15971 98,14243 102,33637 104,3083 106,63667 110,20789 112,24547 114,9059 118,88093 120,11317 122,109619 126,70717 128,46349 130,49891 134,244109 136,70237 138,105691 142,132709 144,18269 146,425387 150,221261 152,266117 154,62323 158,235541 160,31699 162,139907 166,102877 168,65371 170,142211 174,199037 176,265163 178,296299 182,223829 184,411013 186,137723 190,699757 192,191837 194,658643 198,103093 200,339827 202,302227 206,2989757 208,806581 210,425603 214,155797 216,598427 218,184259 222,736469 224,514949 226,2373307 230,1530293 232,1078411 234,512819 238,1427509 240,370133 242,1353371 246,2084653 248,476603 250,2551099 254,4013573 256,2617651 A more generalization: gaps of primes p such that znorder(Mod(b,p)) = (p-1)/a, for fixed integers a>=1, b>=2 (for some (a,b) pairs such primes do not exist, e.g. (4,3) and (5,5)).
 2020-01-19, 14:49 #6 sweety439     Nov 2016 22·13·37 Posts This is a list of the primes p such that p+2 is also prime, and both p and p+2 have primitive root 2: 3, 11, 59, 179, 347, 419, 659, 827, 1451, 1619, 1667, 2027, 2267, 3467, 3851, 4019, 4091, 4259, 4787, 6779, 6827, 6947, 7547, 8219, 8291, 8819, 9419, 10067, 10091, 10139, 10499, 10859, 12251, 12611, 13931, 14387, 14627, 14867, 16067, 16187, 16979, 17387, 17747, 19139, 20507, 20771, 21011, 21491, 21587, 21611, 22619, 22859, 23027, 23627, 24107, 25931, 27059, 28307, 28547, 28571, 29387, 30011, 30467, 30851, 32027, 32531, 32939, 33347, 33827, 34211, 34259, 35051, 35507, 36011, 36107, 36467, 36779, 36899, 37547, 38651, 38747, 39227, 44267, 44531, 44699, 45587, 46091, 46307, 47147, 47387, 47699, 48539, 49331, 49667, 49739, 50051, 50891, 51059, 51419, 51827, 51971, 52067, 54419, 54539, 55619, 56267, 56891, 57347, 57899, 58787, 58907, 59219, 59627, 60659, 60899, 61331, 62987, 63419, 63587, 64187, 64451, 65027, 65099, 65171, 65267, ... Are there infinitely many such primes?
2020-01-19, 14:58   #7
sweety439

Nov 2016

22×13×37 Posts

Quote:
 Originally Posted by sweety439 For gaps between primes p which 2 is primitive root mod p: A more generalization: gaps of primes p such that znorder(Mod(b,p)) = (p-1)/a, for fixed integers a>=1, b>=2 (for some (a,b) pairs such primes do not exist, e.g. (4,3) and (5,5)).
For the smallest prime p such that znorder(Mod(m,p)) = (p-1)/n, for fixed integers 2<=m<=32, 1<=n<=32 (0 if not exist):

Code:
m\n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
2: 3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593,
3: 2, 11, 67, 13, 41, 61, 883, 313, 271, 431, 5743, 193, 3511, 1583, 2131, 433, 2551, 4177, 8513, 2521, 8779, 683, 10627, 1321, 29851, 1223, 3079, 9661, 49939, 661, 101681, 4129,
4: 0, 3, 0, 17, 0, 31, 0, 73, 0, 151, 0, 433, 0, 631, 0, 337, 0, 127, 0, 241, 0, 331, 0, 601, 0, 4421, 0, 673, 0, 3061, 0, 257,
5: 2, 11, 13, 101, 0, 199, 827, 569, 487, 31, 1453, 181, 7853, 71, 0, 401, 5407, 379, 15277, 761, 1303, 2069, 5107, 409, 0, 1171, 5077, 3109, 1973, 2521, 5023, 449,
6: 11, 19, 7, 5, 31, 139, 463, 97, 37, 101, 353, 241, 4889, 43, 421, 5233, 6563, 1747, 8171, 1901, 11551, 3719, 3037, 409, 28001, 26833, 26407, 11789, 5801, 3931, 48299, 15073,
7: 2, 3, 73, 29, 1031, 19, 43, 113, 883, 311, 353, 1453, 2861, 281, 181, 1873, 409, 1531, 191, 1621, 2311, 419, 14629, 5233, 12251, 7333, 32941, 4397, 11717, 811, 23251, 1409,
8: 3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641,
9: 0, 5, 0, 13, 0, 67, 0, 313, 0, 41, 0, 61, 0, 883, 0, 433, 0, 271, 0, 2161, 0, 683, 0, 193, 0, 1223, 0, 8317, 0, 2131, 0, 769,
10: 7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289,
11: 2, 7, 193, 5, 191, 19, 379, 449, 199, 1301, 2531, 1549, 2081, 547, 61, 1697, 2789, 523, 28843, 661, 1303, 1013, 18539, 2377, 4001, 1847, 31267, 6917, 10499, 1231, 39929, 6689,
12: 5, 23, 19, 37, 271, 13, 29, 193, 487, 11, 89, 373, 521, 421, 211, 5521, 7243, 829, 2129, 1741, 20707, 1453, 10903, 673, 17551, 4993, 12799, 5209, 233, 3181, 25793, 3169,
13: 2, 3, 7, 17, 331, 103, 2017, 673, 1657, 311, 463, 1213, 0, 1303, 271, 337, 1123, 1171, 19001, 61, 421, 7283, 4049, 2617, 1151, 157, 3889, 701, 8237, 601, 71983, 641,
14: 3, 5, 37, 113, 41, 67, 71, 401, 1459, 61, 463, 13, 3121, 659, 1381, 977, 41413, 1009, 1597, 461, 967, 8779, 23369, 12049, 9151, 547, 811, 8233, 132299, 5431, 148367, 2081,
15: 2, 11, 31, 53, 761, 7, 1163, 257, 3691, 311, 991, 1549, 443, 617, 2551, 2417, 1361, 1801, 2129, 3541, 3697, 1123, 12329, 5641, 4651, 2393, 4159, 113, 9629, 1201, 23003, 1249,
16: 0, 3, 0, 5, 0, 31, 0, 17, 0, 151, 0, 109, 0, 631, 0, 113, 0, 127, 0, 1181, 0, 331, 0, 433, 0, 13963, 0, 1709, 0, 3331, 0, 1217,
17: 2, 13, 73, 149, 181, 223, 29, 257, 541, 101, 2003, 229, 1093, 1471, 991, 433, 0, 883, 2851, 1361, 3361, 1409, 19183, 3673, 13901, 3719, 7723, 8093, 6091, 2371, 10789, 1889,
18: 5, 7, 13, 73, 131, 79, 1667, 41, 19, 311, 3917, 1201, 443, 113, 1381, 17, 1259, 199, 229, 2801, 1429, 881, 1427, 1153, 18701, 599, 12853, 6833, 20939, 2671, 19469, 3361,
19: 2, 3, 97, 101, 131, 307, 1303, 233, 271, 1291, 199, 277, 859, 197, 691, 1217, 12037, 487, 24967, 1901, 1009, 8999, 2393, 4561, 4951, 5227, 6373, 8513, 56957, 151, 14447, 2753,
20: 3, 11, 7, 29, 0, 151, 197, 521, 577, 71, 617, 61, 1873, 491, 0, 1489, 307, 19, 7753, 661, 127, 4049, 9293, 1129, 0, 859, 3673, 3221, 44777, 691, 8123, 929,
21: 2, 37, 13, 5, 11, 43, 953, 337, 433, 461, 199, 1129, 599, 211, 661, 881, 3877, 1747, 14897, 3301, 0, 1277, 52901, 1801, 14551, 30707, 2971, 14197, 34337, 1171, 41231, 1697,
22: 5, 3, 43, 13, 241, 7, 631, 521, 73, 461, 23, 613, 157, 127, 5791, 433, 10337, 2647, 37013, 401, 4201, 947, 17021, 97, 12101, 3407, 15013, 6329, 14153, 1381, 12959, 353,
23: 2, 7, 31, 29, 71, 103, 239, 233, 163, 11, 859, 1093, 53, 911, 271, 1153, 7039, 2719, 25423, 461, 211, 1013, 5843, 3889, 1901, 79, 57349, 1933, 13399, 2131, 17299, 4129,
24: 7, 5, 61, 29, 131, 67, 127, 457, 613, 311, 199, 2617, 79, 379, 991, 241, 4999, 307, 12541, 6581, 8527, 23, 11777, 1009, 1451, 4967, 22303, 2381, 349, 1321, 5023, 4801,
25: 0, 3, 0, 29, 0, 13, 0, 569, 0, 31, 0, 181, 0, 71, 0, 401, 0, 379, 0, 641, 0, 1453, 0, 409, 0, 1171, 0, 3109, 0, 2851, 0, 8609,
26: 3, 11, 151, 5, 31, 19, 547, 313, 1657, 1031, 859, 37, 6397, 3823, 181, 337, 4421, 3853, 4409, 7741, 757, 2311, 37307, 8161, 3701, 2393, 19441, 1597, 1567, 5101, 23561, 4001,
27: 2, 11, 7, 0, 41, 37, 1289, 0, 307, 431, 9857, 13, 7853, 1583, 1051, 0, 7481, 73, 8513, 0, 883, 683, 14813, 313, 38501, 1223, 271, 0, 59393, 661, 101681, 0,
28: 5, 3, 61, 53, 601, 199, 127, 449, 1423, 281, 4093, 1117, 3719, 29, 631, 113, 4999, 613, 23447, 541, 547, 6359, 6211, 6073, 14851, 4733, 4159, 6469, 33641, 4561, 1861, 6113,
29: 2, 5, 31, 13, 61, 7, 617, 1289, 541, 571, 727, 181, 2549, 673, 3121, 2609, 1259, 3061, 2927, 11981, 757, 67, 12743, 7321, 11701, 313, 16417, 12853, 0, 1831, 8123, 12577,
30: 11, 7, 73, 17, 991, 19, 1289, 257, 163, 71, 67, 277, 53, 1163, 31, 113, 1259, 613, 7069, 461, 337, 947, 9293, 409, 401, 1171, 3673, 29, 52259, 241, 14323, 10337,
31: 2, 3, 13, 5, 191, 271, 659, 977, 37, 541, 5237, 349, 4759, 911, 4111, 1217, 2143, 2683, 2129, 3221, 2689, 3499, 2531, 2857, 7901, 1613, 11827, 2437, 45821, 571, 40487, 577,
32: 3, 7, 43, 113, 11, 223, 1163, 73, 397, 41, 1013, 1753, 4733, 673, 691, 257, 1429, 127, 6043, 281, 33013, 6337, 18539, 1777, 251, 14327, 5347, 2857, 72269, 31, 683, 2593,
For more values (2<=m<=64, 1<=n<=64), see https://de.wikipedia.org/w/index.php...ldid=195976169

Last fiddled with by sweety439 on 2020-01-19 at 15:13

2020-01-19, 15:35   #8
sweety439

Nov 2016

192410 Posts

Quote:
 Originally Posted by sweety439 For the smallest prime p such that znorder(Mod(m,p)) = (p-1)/n, for fixed integers 2<=m<=32, 1<=n<=32 (0 if not exist): Code: m\n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 2: 3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 3: 2, 11, 67, 13, 41, 61, 883, 313, 271, 431, 5743, 193, 3511, 1583, 2131, 433, 2551, 4177, 8513, 2521, 8779, 683, 10627, 1321, 29851, 1223, 3079, 9661, 49939, 661, 101681, 4129, 4: 0, 3, 0, 17, 0, 31, 0, 73, 0, 151, 0, 433, 0, 631, 0, 337, 0, 127, 0, 241, 0, 331, 0, 601, 0, 4421, 0, 673, 0, 3061, 0, 257, 5: 2, 11, 13, 101, 0, 199, 827, 569, 487, 31, 1453, 181, 7853, 71, 0, 401, 5407, 379, 15277, 761, 1303, 2069, 5107, 409, 0, 1171, 5077, 3109, 1973, 2521, 5023, 449, 6: 11, 19, 7, 5, 31, 139, 463, 97, 37, 101, 353, 241, 4889, 43, 421, 5233, 6563, 1747, 8171, 1901, 11551, 3719, 3037, 409, 28001, 26833, 26407, 11789, 5801, 3931, 48299, 15073, 7: 2, 3, 73, 29, 1031, 19, 43, 113, 883, 311, 353, 1453, 2861, 281, 181, 1873, 409, 1531, 191, 1621, 2311, 419, 14629, 5233, 12251, 7333, 32941, 4397, 11717, 811, 23251, 1409, 8: 3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641, 9: 0, 5, 0, 13, 0, 67, 0, 313, 0, 41, 0, 61, 0, 883, 0, 433, 0, 271, 0, 2161, 0, 683, 0, 193, 0, 1223, 0, 8317, 0, 2131, 0, 769, 10: 7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 11: 2, 7, 193, 5, 191, 19, 379, 449, 199, 1301, 2531, 1549, 2081, 547, 61, 1697, 2789, 523, 28843, 661, 1303, 1013, 18539, 2377, 4001, 1847, 31267, 6917, 10499, 1231, 39929, 6689, 12: 5, 23, 19, 37, 271, 13, 29, 193, 487, 11, 89, 373, 521, 421, 211, 5521, 7243, 829, 2129, 1741, 20707, 1453, 10903, 673, 17551, 4993, 12799, 5209, 233, 3181, 25793, 3169, 13: 2, 3, 7, 17, 331, 103, 2017, 673, 1657, 311, 463, 1213, 0, 1303, 271, 337, 1123, 1171, 19001, 61, 421, 7283, 4049, 2617, 1151, 157, 3889, 701, 8237, 601, 71983, 641, 14: 3, 5, 37, 113, 41, 67, 71, 401, 1459, 61, 463, 13, 3121, 659, 1381, 977, 41413, 1009, 1597, 461, 967, 8779, 23369, 12049, 9151, 547, 811, 8233, 132299, 5431, 148367, 2081, 15: 2, 11, 31, 53, 761, 7, 1163, 257, 3691, 311, 991, 1549, 443, 617, 2551, 2417, 1361, 1801, 2129, 3541, 3697, 1123, 12329, 5641, 4651, 2393, 4159, 113, 9629, 1201, 23003, 1249, 16: 0, 3, 0, 5, 0, 31, 0, 17, 0, 151, 0, 109, 0, 631, 0, 113, 0, 127, 0, 1181, 0, 331, 0, 433, 0, 13963, 0, 1709, 0, 3331, 0, 1217, 17: 2, 13, 73, 149, 181, 223, 29, 257, 541, 101, 2003, 229, 1093, 1471, 991, 433, 0, 883, 2851, 1361, 3361, 1409, 19183, 3673, 13901, 3719, 7723, 8093, 6091, 2371, 10789, 1889, 18: 5, 7, 13, 73, 131, 79, 1667, 41, 19, 311, 3917, 1201, 443, 113, 1381, 17, 1259, 199, 229, 2801, 1429, 881, 1427, 1153, 18701, 599, 12853, 6833, 20939, 2671, 19469, 3361, 19: 2, 3, 97, 101, 131, 307, 1303, 233, 271, 1291, 199, 277, 859, 197, 691, 1217, 12037, 487, 24967, 1901, 1009, 8999, 2393, 4561, 4951, 5227, 6373, 8513, 56957, 151, 14447, 2753, 20: 3, 11, 7, 29, 0, 151, 197, 521, 577, 71, 617, 61, 1873, 491, 0, 1489, 307, 19, 7753, 661, 127, 4049, 9293, 1129, 0, 859, 3673, 3221, 44777, 691, 8123, 929, 21: 2, 37, 13, 5, 11, 43, 953, 337, 433, 461, 199, 1129, 599, 211, 661, 881, 3877, 1747, 14897, 3301, 0, 1277, 52901, 1801, 14551, 30707, 2971, 14197, 34337, 1171, 41231, 1697, 22: 5, 3, 43, 13, 241, 7, 631, 521, 73, 461, 23, 613, 157, 127, 5791, 433, 10337, 2647, 37013, 401, 4201, 947, 17021, 97, 12101, 3407, 15013, 6329, 14153, 1381, 12959, 353, 23: 2, 7, 31, 29, 71, 103, 239, 233, 163, 11, 859, 1093, 53, 911, 271, 1153, 7039, 2719, 25423, 461, 211, 1013, 5843, 3889, 1901, 79, 57349, 1933, 13399, 2131, 17299, 4129, 24: 7, 5, 61, 29, 131, 67, 127, 457, 613, 311, 199, 2617, 79, 379, 991, 241, 4999, 307, 12541, 6581, 8527, 23, 11777, 1009, 1451, 4967, 22303, 2381, 349, 1321, 5023, 4801, 25: 0, 3, 0, 29, 0, 13, 0, 569, 0, 31, 0, 181, 0, 71, 0, 401, 0, 379, 0, 641, 0, 1453, 0, 409, 0, 1171, 0, 3109, 0, 2851, 0, 8609, 26: 3, 11, 151, 5, 31, 19, 547, 313, 1657, 1031, 859, 37, 6397, 3823, 181, 337, 4421, 3853, 4409, 7741, 757, 2311, 37307, 8161, 3701, 2393, 19441, 1597, 1567, 5101, 23561, 4001, 27: 2, 11, 7, 0, 41, 37, 1289, 0, 307, 431, 9857, 13, 7853, 1583, 1051, 0, 7481, 73, 8513, 0, 883, 683, 14813, 313, 38501, 1223, 271, 0, 59393, 661, 101681, 0, 28: 5, 3, 61, 53, 601, 199, 127, 449, 1423, 281, 4093, 1117, 3719, 29, 631, 113, 4999, 613, 23447, 541, 547, 6359, 6211, 6073, 14851, 4733, 4159, 6469, 33641, 4561, 1861, 6113, 29: 2, 5, 31, 13, 61, 7, 617, 1289, 541, 571, 727, 181, 2549, 673, 3121, 2609, 1259, 3061, 2927, 11981, 757, 67, 12743, 7321, 11701, 313, 16417, 12853, 0, 1831, 8123, 12577, 30: 11, 7, 73, 17, 991, 19, 1289, 257, 163, 71, 67, 277, 53, 1163, 31, 113, 1259, 613, 7069, 461, 337, 947, 9293, 409, 401, 1171, 3673, 29, 52259, 241, 14323, 10337, 31: 2, 3, 13, 5, 191, 271, 659, 977, 37, 541, 5237, 349, 4759, 911, 4111, 1217, 2143, 2683, 2129, 3221, 2689, 3499, 2531, 2857, 7901, 1613, 11827, 2437, 45821, 571, 40487, 577, 32: 3, 7, 43, 113, 11, 223, 1163, 73, 397, 41, 1013, 1753, 4733, 673, 691, 257, 1429, 127, 6043, 281, 33013, 6337, 18539, 1777, 251, 14327, 5347, 2857, 72269, 31, 683, 2593,
Also see https://de.wikipedia.org/w/index.php...ldid=195978132 for 2<=m<=128, 1<=n<=128 (0 if not exist). (I searched up to p=2^24, and I assume that there are no p>2^24 which is the smallest prime such that znorder(Mod(m,p)) = (p-1)/n for some m,n <= 128, in the list, the largest such prime is 2334251 (for m=126, n=125), and 2334251 is just between 2^21 and 2^22)

Last fiddled with by sweety439 on 2020-01-19 at 16:11

2020-01-19, 16:12   #9
sweety439

Nov 2016

22·13·37 Posts

Quote:
 Originally Posted by sweety439 For the smallest prime p such that znorder(Mod(m,p)) = (p-1)/n, for fixed integers 2<=m<=32, 1<=n<=32 (0 if not exist): Code: m\n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 2: 3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 3: 2, 11, 67, 13, 41, 61, 883, 313, 271, 431, 5743, 193, 3511, 1583, 2131, 433, 2551, 4177, 8513, 2521, 8779, 683, 10627, 1321, 29851, 1223, 3079, 9661, 49939, 661, 101681, 4129, 4: 0, 3, 0, 17, 0, 31, 0, 73, 0, 151, 0, 433, 0, 631, 0, 337, 0, 127, 0, 241, 0, 331, 0, 601, 0, 4421, 0, 673, 0, 3061, 0, 257, 5: 2, 11, 13, 101, 0, 199, 827, 569, 487, 31, 1453, 181, 7853, 71, 0, 401, 5407, 379, 15277, 761, 1303, 2069, 5107, 409, 0, 1171, 5077, 3109, 1973, 2521, 5023, 449, 6: 11, 19, 7, 5, 31, 139, 463, 97, 37, 101, 353, 241, 4889, 43, 421, 5233, 6563, 1747, 8171, 1901, 11551, 3719, 3037, 409, 28001, 26833, 26407, 11789, 5801, 3931, 48299, 15073, 7: 2, 3, 73, 29, 1031, 19, 43, 113, 883, 311, 353, 1453, 2861, 281, 181, 1873, 409, 1531, 191, 1621, 2311, 419, 14629, 5233, 12251, 7333, 32941, 4397, 11717, 811, 23251, 1409, 8: 3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641, 9: 0, 5, 0, 13, 0, 67, 0, 313, 0, 41, 0, 61, 0, 883, 0, 433, 0, 271, 0, 2161, 0, 683, 0, 193, 0, 1223, 0, 8317, 0, 2131, 0, 769, 10: 7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 11: 2, 7, 193, 5, 191, 19, 379, 449, 199, 1301, 2531, 1549, 2081, 547, 61, 1697, 2789, 523, 28843, 661, 1303, 1013, 18539, 2377, 4001, 1847, 31267, 6917, 10499, 1231, 39929, 6689, 12: 5, 23, 19, 37, 271, 13, 29, 193, 487, 11, 89, 373, 521, 421, 211, 5521, 7243, 829, 2129, 1741, 20707, 1453, 10903, 673, 17551, 4993, 12799, 5209, 233, 3181, 25793, 3169, 13: 2, 3, 7, 17, 331, 103, 2017, 673, 1657, 311, 463, 1213, 0, 1303, 271, 337, 1123, 1171, 19001, 61, 421, 7283, 4049, 2617, 1151, 157, 3889, 701, 8237, 601, 71983, 641, 14: 3, 5, 37, 113, 41, 67, 71, 401, 1459, 61, 463, 13, 3121, 659, 1381, 977, 41413, 1009, 1597, 461, 967, 8779, 23369, 12049, 9151, 547, 811, 8233, 132299, 5431, 148367, 2081, 15: 2, 11, 31, 53, 761, 7, 1163, 257, 3691, 311, 991, 1549, 443, 617, 2551, 2417, 1361, 1801, 2129, 3541, 3697, 1123, 12329, 5641, 4651, 2393, 4159, 113, 9629, 1201, 23003, 1249, 16: 0, 3, 0, 5, 0, 31, 0, 17, 0, 151, 0, 109, 0, 631, 0, 113, 0, 127, 0, 1181, 0, 331, 0, 433, 0, 13963, 0, 1709, 0, 3331, 0, 1217, 17: 2, 13, 73, 149, 181, 223, 29, 257, 541, 101, 2003, 229, 1093, 1471, 991, 433, 0, 883, 2851, 1361, 3361, 1409, 19183, 3673, 13901, 3719, 7723, 8093, 6091, 2371, 10789, 1889, 18: 5, 7, 13, 73, 131, 79, 1667, 41, 19, 311, 3917, 1201, 443, 113, 1381, 17, 1259, 199, 229, 2801, 1429, 881, 1427, 1153, 18701, 599, 12853, 6833, 20939, 2671, 19469, 3361, 19: 2, 3, 97, 101, 131, 307, 1303, 233, 271, 1291, 199, 277, 859, 197, 691, 1217, 12037, 487, 24967, 1901, 1009, 8999, 2393, 4561, 4951, 5227, 6373, 8513, 56957, 151, 14447, 2753, 20: 3, 11, 7, 29, 0, 151, 197, 521, 577, 71, 617, 61, 1873, 491, 0, 1489, 307, 19, 7753, 661, 127, 4049, 9293, 1129, 0, 859, 3673, 3221, 44777, 691, 8123, 929, 21: 2, 37, 13, 5, 11, 43, 953, 337, 433, 461, 199, 1129, 599, 211, 661, 881, 3877, 1747, 14897, 3301, 0, 1277, 52901, 1801, 14551, 30707, 2971, 14197, 34337, 1171, 41231, 1697, 22: 5, 3, 43, 13, 241, 7, 631, 521, 73, 461, 23, 613, 157, 127, 5791, 433, 10337, 2647, 37013, 401, 4201, 947, 17021, 97, 12101, 3407, 15013, 6329, 14153, 1381, 12959, 353, 23: 2, 7, 31, 29, 71, 103, 239, 233, 163, 11, 859, 1093, 53, 911, 271, 1153, 7039, 2719, 25423, 461, 211, 1013, 5843, 3889, 1901, 79, 57349, 1933, 13399, 2131, 17299, 4129, 24: 7, 5, 61, 29, 131, 67, 127, 457, 613, 311, 199, 2617, 79, 379, 991, 241, 4999, 307, 12541, 6581, 8527, 23, 11777, 1009, 1451, 4967, 22303, 2381, 349, 1321, 5023, 4801, 25: 0, 3, 0, 29, 0, 13, 0, 569, 0, 31, 0, 181, 0, 71, 0, 401, 0, 379, 0, 641, 0, 1453, 0, 409, 0, 1171, 0, 3109, 0, 2851, 0, 8609, 26: 3, 11, 151, 5, 31, 19, 547, 313, 1657, 1031, 859, 37, 6397, 3823, 181, 337, 4421, 3853, 4409, 7741, 757, 2311, 37307, 8161, 3701, 2393, 19441, 1597, 1567, 5101, 23561, 4001, 27: 2, 11, 7, 0, 41, 37, 1289, 0, 307, 431, 9857, 13, 7853, 1583, 1051, 0, 7481, 73, 8513, 0, 883, 683, 14813, 313, 38501, 1223, 271, 0, 59393, 661, 101681, 0, 28: 5, 3, 61, 53, 601, 199, 127, 449, 1423, 281, 4093, 1117, 3719, 29, 631, 113, 4999, 613, 23447, 541, 547, 6359, 6211, 6073, 14851, 4733, 4159, 6469, 33641, 4561, 1861, 6113, 29: 2, 5, 31, 13, 61, 7, 617, 1289, 541, 571, 727, 181, 2549, 673, 3121, 2609, 1259, 3061, 2927, 11981, 757, 67, 12743, 7321, 11701, 313, 16417, 12853, 0, 1831, 8123, 12577, 30: 11, 7, 73, 17, 991, 19, 1289, 257, 163, 71, 67, 277, 53, 1163, 31, 113, 1259, 613, 7069, 461, 337, 947, 9293, 409, 401, 1171, 3673, 29, 52259, 241, 14323, 10337, 31: 2, 3, 13, 5, 191, 271, 659, 977, 37, 541, 5237, 349, 4759, 911, 4111, 1217, 2143, 2683, 2129, 3221, 2689, 3499, 2531, 2857, 7901, 1613, 11827, 2437, 45821, 571, 40487, 577, 32: 3, 7, 43, 113, 11, 223, 1163, 73, 397, 41, 1013, 1753, 4733, 673, 691, 257, 1429, 127, 6043, 281, 33013, 6337, 18539, 1777, 251, 14327, 5347, 2857, 72269, 31, 683, 2593, For more values (2<=m<=64, 1<=n<=64), see https://de.wikipedia.org/w/index.php...ldid=195976169
The p for (m,n) = (9,1) and (25,1) are 2, not 0

2020-01-19, 22:20   #10
sweety439

Nov 2016

192410 Posts

Quote:
 Originally Posted by sweety439 For the smallest prime p such that znorder(Mod(m,p)) = (p-1)/n, for fixed integers 2<=m<=32, 1<=n<=32 (0 if not exist): Code: m\n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 2: 3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 3: 2, 11, 67, 13, 41, 61, 883, 313, 271, 431, 5743, 193, 3511, 1583, 2131, 433, 2551, 4177, 8513, 2521, 8779, 683, 10627, 1321, 29851, 1223, 3079, 9661, 49939, 661, 101681, 4129, 4: 0, 3, 0, 17, 0, 31, 0, 73, 0, 151, 0, 433, 0, 631, 0, 337, 0, 127, 0, 241, 0, 331, 0, 601, 0, 4421, 0, 673, 0, 3061, 0, 257, 5: 2, 11, 13, 101, 0, 199, 827, 569, 487, 31, 1453, 181, 7853, 71, 0, 401, 5407, 379, 15277, 761, 1303, 2069, 5107, 409, 0, 1171, 5077, 3109, 1973, 2521, 5023, 449, 6: 11, 19, 7, 5, 31, 139, 463, 97, 37, 101, 353, 241, 4889, 43, 421, 5233, 6563, 1747, 8171, 1901, 11551, 3719, 3037, 409, 28001, 26833, 26407, 11789, 5801, 3931, 48299, 15073, 7: 2, 3, 73, 29, 1031, 19, 43, 113, 883, 311, 353, 1453, 2861, 281, 181, 1873, 409, 1531, 191, 1621, 2311, 419, 14629, 5233, 12251, 7333, 32941, 4397, 11717, 811, 23251, 1409, 8: 3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641, 9: 0, 5, 0, 13, 0, 67, 0, 313, 0, 41, 0, 61, 0, 883, 0, 433, 0, 271, 0, 2161, 0, 683, 0, 193, 0, 1223, 0, 8317, 0, 2131, 0, 769, 10: 7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 11: 2, 7, 193, 5, 191, 19, 379, 449, 199, 1301, 2531, 1549, 2081, 547, 61, 1697, 2789, 523, 28843, 661, 1303, 1013, 18539, 2377, 4001, 1847, 31267, 6917, 10499, 1231, 39929, 6689, 12: 5, 23, 19, 37, 271, 13, 29, 193, 487, 11, 89, 373, 521, 421, 211, 5521, 7243, 829, 2129, 1741, 20707, 1453, 10903, 673, 17551, 4993, 12799, 5209, 233, 3181, 25793, 3169, 13: 2, 3, 7, 17, 331, 103, 2017, 673, 1657, 311, 463, 1213, 0, 1303, 271, 337, 1123, 1171, 19001, 61, 421, 7283, 4049, 2617, 1151, 157, 3889, 701, 8237, 601, 71983, 641, 14: 3, 5, 37, 113, 41, 67, 71, 401, 1459, 61, 463, 13, 3121, 659, 1381, 977, 41413, 1009, 1597, 461, 967, 8779, 23369, 12049, 9151, 547, 811, 8233, 132299, 5431, 148367, 2081, 15: 2, 11, 31, 53, 761, 7, 1163, 257, 3691, 311, 991, 1549, 443, 617, 2551, 2417, 1361, 1801, 2129, 3541, 3697, 1123, 12329, 5641, 4651, 2393, 4159, 113, 9629, 1201, 23003, 1249, 16: 0, 3, 0, 5, 0, 31, 0, 17, 0, 151, 0, 109, 0, 631, 0, 113, 0, 127, 0, 1181, 0, 331, 0, 433, 0, 13963, 0, 1709, 0, 3331, 0, 1217, 17: 2, 13, 73, 149, 181, 223, 29, 257, 541, 101, 2003, 229, 1093, 1471, 991, 433, 0, 883, 2851, 1361, 3361, 1409, 19183, 3673, 13901, 3719, 7723, 8093, 6091, 2371, 10789, 1889, 18: 5, 7, 13, 73, 131, 79, 1667, 41, 19, 311, 3917, 1201, 443, 113, 1381, 17, 1259, 199, 229, 2801, 1429, 881, 1427, 1153, 18701, 599, 12853, 6833, 20939, 2671, 19469, 3361, 19: 2, 3, 97, 101, 131, 307, 1303, 233, 271, 1291, 199, 277, 859, 197, 691, 1217, 12037, 487, 24967, 1901, 1009, 8999, 2393, 4561, 4951, 5227, 6373, 8513, 56957, 151, 14447, 2753, 20: 3, 11, 7, 29, 0, 151, 197, 521, 577, 71, 617, 61, 1873, 491, 0, 1489, 307, 19, 7753, 661, 127, 4049, 9293, 1129, 0, 859, 3673, 3221, 44777, 691, 8123, 929, 21: 2, 37, 13, 5, 11, 43, 953, 337, 433, 461, 199, 1129, 599, 211, 661, 881, 3877, 1747, 14897, 3301, 0, 1277, 52901, 1801, 14551, 30707, 2971, 14197, 34337, 1171, 41231, 1697, 22: 5, 3, 43, 13, 241, 7, 631, 521, 73, 461, 23, 613, 157, 127, 5791, 433, 10337, 2647, 37013, 401, 4201, 947, 17021, 97, 12101, 3407, 15013, 6329, 14153, 1381, 12959, 353, 23: 2, 7, 31, 29, 71, 103, 239, 233, 163, 11, 859, 1093, 53, 911, 271, 1153, 7039, 2719, 25423, 461, 211, 1013, 5843, 3889, 1901, 79, 57349, 1933, 13399, 2131, 17299, 4129, 24: 7, 5, 61, 29, 131, 67, 127, 457, 613, 311, 199, 2617, 79, 379, 991, 241, 4999, 307, 12541, 6581, 8527, 23, 11777, 1009, 1451, 4967, 22303, 2381, 349, 1321, 5023, 4801, 25: 0, 3, 0, 29, 0, 13, 0, 569, 0, 31, 0, 181, 0, 71, 0, 401, 0, 379, 0, 641, 0, 1453, 0, 409, 0, 1171, 0, 3109, 0, 2851, 0, 8609, 26: 3, 11, 151, 5, 31, 19, 547, 313, 1657, 1031, 859, 37, 6397, 3823, 181, 337, 4421, 3853, 4409, 7741, 757, 2311, 37307, 8161, 3701, 2393, 19441, 1597, 1567, 5101, 23561, 4001, 27: 2, 11, 7, 0, 41, 37, 1289, 0, 307, 431, 9857, 13, 7853, 1583, 1051, 0, 7481, 73, 8513, 0, 883, 683, 14813, 313, 38501, 1223, 271, 0, 59393, 661, 101681, 0, 28: 5, 3, 61, 53, 601, 199, 127, 449, 1423, 281, 4093, 1117, 3719, 29, 631, 113, 4999, 613, 23447, 541, 547, 6359, 6211, 6073, 14851, 4733, 4159, 6469, 33641, 4561, 1861, 6113, 29: 2, 5, 31, 13, 61, 7, 617, 1289, 541, 571, 727, 181, 2549, 673, 3121, 2609, 1259, 3061, 2927, 11981, 757, 67, 12743, 7321, 11701, 313, 16417, 12853, 0, 1831, 8123, 12577, 30: 11, 7, 73, 17, 991, 19, 1289, 257, 163, 71, 67, 277, 53, 1163, 31, 113, 1259, 613, 7069, 461, 337, 947, 9293, 409, 401, 1171, 3673, 29, 52259, 241, 14323, 10337, 31: 2, 3, 13, 5, 191, 271, 659, 977, 37, 541, 5237, 349, 4759, 911, 4111, 1217, 2143, 2683, 2129, 3221, 2689, 3499, 2531, 2857, 7901, 1613, 11827, 2437, 45821, 571, 40487, 577, 32: 3, 7, 43, 113, 11, 223, 1163, 73, 397, 41, 1013, 1753, 4733, 673, 691, 257, 1429, 127, 6043, 281, 33013, 6337, 18539, 1777, 251, 14327, 5347, 2857, 72269, 31, 683, 2593, For more values (2<=m<=64, 1<=n<=64), see https://de.wikipedia.org/w/index.php...ldid=195976169
If n == 2 mod 4, then such prime exists for every m>=2

If n is divisible by 12, then such prime exists for every m>=2

Such prime does not exist if and only if one of these three conditions holds: (note that any pair of two of these three conditions cannot both hold)

* m is square, n is odd, n > 1

* Let m' be the squarefree part of m, m' == 1 mod 4, m' > 1, n is odd, and n is multiple of m'

* m is of the form 27*k^6, n == 4 or 8 mod 12

Code:
m    n such that such prime does not exist
4    m == 1 mod 2
5    m == 5 mod 10
9    m == 1 mod 2 (except m = 1)
13   m == 13 mod 26
16   m == 1 mod 2
17   m == 17 mod 34
20   m == 5 mod 10
21   m == 21 mod 42
25   m == 1 mod 2 (except m = 1)
27   m == 4, 8 mod 12
29   m == 29 mod 58
33   m == 33 mod 66
36   m == 1 mod 2
37   m == 37 mod 74
41   m == 41 mod 82
45   m == 5 mod 10
49   m == 1 mod 2 (except m = 1)
52   m == 13 mod 26
53   m == 53 mod 106
57   m == 57 mod 114
61   m == 61 mod 122
64   m == 1 mod 2
65   m == 65 mod 130
68   m == 17 mod 34
69   m == 69 mod 138
73   m == 73 mod 146
77   m == 77 mod 154
80   m == 5 mod 10
81   m == 1 mod 2 (except m = 1)
84   m == 21 mod 42
85   m == 85 mod 170
89   m == 89 mod 178
93   m == 93 mod 186
97   m == 97 mod 194
100  m == 1 mod 2
101  m == 101 mod 202
105  m == 105 mod 210
109  m == 109 mod 218
113  m == 113 mod 226
116  m == 29 mod 58
117  m == 13 mod 26
121  m == 1 mod 2 (except m = 1)
125  m == 5 mod 10

Last fiddled with by sweety439 on 2020-01-19 at 22:35

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