mersenneforum.org (https://www.mersenneforum.org/index.php)
-   Miscellaneous Math (https://www.mersenneforum.org/forumdisplay.php?f=56)
-   -   Happy New Year & There are just not enough Numerological Threads (https://www.mersenneforum.org/showthread.php?t=26364)

 a1call 2021-01-01 22:31

Happy New Year & There are just not enough Numerological Threads

[CODE]

\\\\DSN-320-A-Pari-GP Code By Rashid Naimi 1/1/2121 >>>>>>>>>>>>>>>Error caught by Krisel. Now I have to reweight all the post dated checks for the year

primeCounter=0
compositeCounter=0
LowP=101
highP=-1
for(n=1,19^19,{\\\\\\\\\\\\\\\\\\\\\\\\\\
n1=2*n;
n2=(2+n*2)*n/2;
m0=3;
lowerFactor1=(p-1)/2;
upperFactor1 = lowerFactor1+1;
mLower1 = Mod(lowerFactor1,p);
mLower1 = mLower1 *lowerFactor1;
mUpper1 = Mod(upperFactor1,p);
mUpper1 = mUpper1*upperFactor1;
lowerFactor1=lowerFactor1+1;
upperFactor1 = upperFactor1-1;
primeFlag=1;
theDepth=1+2^14; \\\\\\ the larger this value, the less composites will pass the test.
if(theDepth>(p-1)/2,theDepth=(p-1)/2);
for(j=1,theDepth,
lowerFactor1=lowerFactor1-1;
upperFactor1 = upperFactor1+1;
\\mLower1 = mLower1/lowerFactor1;
iferr(mLower1 = mLower1/lowerFactor1, E, mLower1 =0;primeFlag=0;);
\\mUpper1 = mUpper1/upperFactor1;
iferr(mUpper1 = mUpper1/upperFactor1, E, mUpper1 =0;primeFlag=0;);
);
if(!primeFlag,
next(1);
,
print("\nn = ",n);
print("n1 = ",n1);
print("n2 = ",n2);
print("p = ",p," >> ", ispseudoprime(p));
print("**** ",LowP,"% < ",prcntg,"% < ",highP,"% Prime!");
);
if(!ispseudoprime(p),
if(primeFlag,
\\next(19); \\Uncomment this line to abort on 1st test failiure
);
);
if(ispseudoprime(p),primeCounter=primeCounter+1,if(primeFlag,compositeCounter=compositeCounter+1));
prcntg =round(primeCounter/(primeCounter+compositeCounter)*100);
if(prcntg < LowP,LowP=prcntg);
if(prcntg > highP,highP=prcntg);
})

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\^^^^^^^^^^^^^^^^^^

[/CODE]

I will let you figure out why it does what it does.:smile:
I am only off by 100 years.:picard:

ETA: Hint #1: It is based on Wilson's Theorem:

[url]https://en.wikipedia.org/wiki/Wilson%27s_theorem[/url]

 a1call 2021-01-02 00:48

Hint #2:
For all positive integers p where valuation(p-1,2)==1
,
Mod( ((p-1)/2)! , p) == 1 || Mod( ((p-1)/2)! , p-1)

if and only if p is prime

Hint #3:
For all primes p where valuation(p-1,2)>1
,
there exist coprime integers a and b such that a+b=p && p | ab-1
(Not all but some composites satisfy this condition as well such as 25 where a=7 and b= 18)

ETA: being off by 100 years isn't that bad really, considering that numbers are unlimited and I could potentially be off by an indefinitely larger value.:rolleyes:
ETA II
Hint #4:

For all primes p where valuation(p-1,2)>1
,
There are coprime integers a and b such that
Mod(( (p-1)/2)!, p) == a
and
Mod(( (p-2)! / (p-1)/2)!, p) == b
and a+b=p
and 1<a<b<p

 a1call 2021-01-02 01:56

Re: Hint #4
if m=Mod(a,p)
Unless p is prime, iterations of
m=m/c
where c iterates from (p-1)/2 to 2 or vise versa will definitely fail to evaluate at some point which is what the code checks.:smile:

Thank you for your time.

 ONeil 2021-01-02 05:16

Hey at1call will the above code predict if a large number is prime or composite effectively and fast?

 a1call 2021-01-02 05:29

Nah, unfortunately you will have to set theDepth parameter very high for large numbers which will make the process unusable for larger numbers. That is unless someone can figure out the mechanics of when m/c fails to evaluate and predict when it will happen.
I gave up trying to do that.:smile:

 a1call 2021-01-05 06:41

So it turns out it's just trial factoring.
Mod(a,p)/k will fail to evaluate if k has a prime cofactor with p.:smile:

 a1call 2021-01-12 05:25

Some more insights from the future (just kidding:smile:).
So it turns out that the number of (a, b) pairs for each n is a function of number of prime factors of n.
If n is prime or otherwise a power of a prime there is only one pair of positive integers (a, b) satisfying the condition:

a+b=n
and
n | ab-1
Where valuation(n-1,2) > 1

---------------------
If n has 2 prime factors there will be 2 pairs of (a, b)'s
if n has 3 prime factors there will be 4. pairs of (a, b)'s
....

Furthermore the differences of multipair (a, b)'s from different pairs will have a common factor with n.
so for n =65
(a, b)[SUB]1[/SUB] = ( 8, 57)
(a, b)[SUB]2[/SUB] = (18, 47)
and
gcd(18-8,65) == gcd(47-57,65) = 5
gcd(57-18,65) == gcd(47-8,65) = 13

where
65 = 5*13
[CODE]
\\DSZ-100-A by Rashid Naimi 1/12/2221

forstep(n=3,19^3,2,{
for(a=2,(n-1)/2,
b=n-a;
m=lift(Mod(a*b,n));
if(m==1,
print("\n",n," >> ",factor(n)," >> ",isprime(n));
print(a,", ",b);
);
);
})

[/CODE]

Output:

[CODE]

5 >> Mat([5, 1]) >> 1
2, 3

13 >> Mat([13, 1]) >> 1
5, 8

17 >> Mat([17, 1]) >> 1
4, 13

25 >> Mat([5, 2]) >> 0
7, 18

29 >> Mat([29, 1]) >> 1
12, 17

37 >> Mat([37, 1]) >> 1
6, 31

41 >> Mat([41, 1]) >> 1
9, 32

53 >> Mat([53, 1]) >> 1
23, 30

61 >> Mat([61, 1]) >> 1
11, 50

65 >> [5, 1; 13, 1] >> 0
8, 57

65 >> [5, 1; 13, 1] >> 0
18, 47

73 >> Mat([73, 1]) >> 1
27, 46

85 >> [5, 1; 17, 1] >> 0
13, 72

85 >> [5, 1; 17, 1] >> 0
38, 47

89 >> Mat([89, 1]) >> 1
34, 55

97 >> Mat([97, 1]) >> 1
22, 75

101 >> Mat([101, 1]) >> 1
10, 91

109 >> Mat([109, 1]) >> 1
33, 76

113 >> Mat([113, 1]) >> 1
15, 98

125 >> Mat([5, 3]) >> 0
57, 68

137 >> Mat([137, 1]) >> 1
37, 100

145 >> [5, 1; 29, 1] >> 0
12, 133

145 >> [5, 1; 29, 1] >> 0
17, 128

149 >> Mat([149, 1]) >> 1
44, 105

157 >> Mat([157, 1]) >> 1
28, 129

169 >> Mat([13, 2]) >> 0
70, 99

173 >> Mat([173, 1]) >> 1
80, 93

181 >> Mat([181, 1]) >> 1
19, 162

185 >> [5, 1; 37, 1] >> 0
43, 142

185 >> [5, 1; 37, 1] >> 0
68, 117

193 >> Mat([193, 1]) >> 1
81, 112

197 >> Mat([197, 1]) >> 1
14, 183

205 >> [5, 1; 41, 1] >> 0
32, 173

205 >> [5, 1; 41, 1] >> 0
73, 132

221 >> [13, 1; 17, 1] >> 0
21, 200

221 >> [13, 1; 17, 1] >> 0
47, 174

229 >> Mat([229, 1]) >> 1
107, 122

233 >> Mat([233, 1]) >> 1
89, 144

241 >> Mat([241, 1]) >> 1
64, 177

257 >> Mat([257, 1]) >> 1
16, 241

265 >> [5, 1; 53, 1] >> 0
23, 242

265 >> [5, 1; 53, 1] >> 0
83, 182

269 >> Mat([269, 1]) >> 1
82, 187

277 >> Mat([277, 1]) >> 1
60, 217

281 >> Mat([281, 1]) >> 1
53, 228

289 >> Mat([17, 2]) >> 0
38, 251

293 >> Mat([293, 1]) >> 1
138, 155

305 >> [5, 1; 61, 1] >> 0
72, 233

305 >> [5, 1; 61, 1] >> 0
133, 172

313 >> Mat([313, 1]) >> 1
25, 288

317 >> Mat([317, 1]) >> 1
114, 203

325 >> [5, 2; 13, 1] >> 0
18, 307

325 >> [5, 2; 13, 1] >> 0
57, 268

337 >> Mat([337, 1]) >> 1
148, 189

349 >> Mat([349, 1]) >> 1
136, 213

353 >> Mat([353, 1]) >> 1
42, 311

365 >> [5, 1; 73, 1] >> 0
27, 338

365 >> [5, 1; 73, 1] >> 0
173, 192

373 >> Mat([373, 1]) >> 1
104, 269

377 >> [13, 1; 29, 1] >> 0
70, 307

377 >> [13, 1; 29, 1] >> 0
99, 278

389 >> Mat([389, 1]) >> 1
115, 274

397 >> Mat([397, 1]) >> 1
63, 334

401 >> Mat([401, 1]) >> 1
20, 381

409 >> Mat([409, 1]) >> 1
143, 266

421 >> Mat([421, 1]) >> 1
29, 392

425 >> [5, 2; 17, 1] >> 0
132, 293

425 >> [5, 2; 17, 1] >> 0
157, 268

433 >> Mat([433, 1]) >> 1
179, 254

445 >> [5, 1; 89, 1] >> 0
123, 322

445 >> [5, 1; 89, 1] >> 0
212, 233

449 >> Mat([449, 1]) >> 1
67, 382

457 >> Mat([457, 1]) >> 1
109, 348

461 >> Mat([461, 1]) >> 1
48, 413

481 >> [13, 1; 37, 1] >> 0
31, 450

481 >> [13, 1; 37, 1] >> 0
216, 265

485 >> [5, 1; 97, 1] >> 0
22, 463

485 >> [5, 1; 97, 1] >> 0
172, 313

493 >> [17, 1; 29, 1] >> 0
157, 336

493 >> [17, 1; 29, 1] >> 0
191, 302

505 >> [5, 1; 101, 1] >> 0
192, 313

505 >> [5, 1; 101, 1] >> 0
212, 293

509 >> Mat([509, 1]) >> 1
208, 301

521 >> Mat([521, 1]) >> 1
235, 286

533 >> [13, 1; 41, 1] >> 0
73, 460

533 >> [13, 1; 41, 1] >> 0
255, 278

541 >> Mat([541, 1]) >> 1
52, 489

545 >> [5, 1; 109, 1] >> 0
33, 512

545 >> [5, 1; 109, 1] >> 0
142, 403

557 >> Mat([557, 1]) >> 1
118, 439

565 >> [5, 1; 113, 1] >> 0
98, 467

565 >> [5, 1; 113, 1] >> 0
128, 437

569 >> Mat([569, 1]) >> 1
86, 483

577 >> Mat([577, 1]) >> 1
24, 553

593 >> Mat([593, 1]) >> 1
77, 516

601 >> Mat([601, 1]) >> 1
125, 476

613 >> Mat([613, 1]) >> 1
35, 578

617 >> Mat([617, 1]) >> 1
194, 423

625 >> Mat([5, 4]) >> 0
182, 443

629 >> [17, 1; 37, 1] >> 0
191, 438

629 >> [17, 1; 37, 1] >> 0
302, 327

641 >> Mat([641, 1]) >> 1
154, 487

653 >> Mat([653, 1]) >> 1
149, 504

661 >> Mat([661, 1]) >> 1
106, 555

673 >> Mat([673, 1]) >> 1
58, 615

677 >> Mat([677, 1]) >> 1
26, 651

685 >> [5, 1; 137, 1] >> 0
37, 648

685 >> [5, 1; 137, 1] >> 0
237, 448

689 >> [13, 1; 53, 1] >> 0
83, 606

689 >> [13, 1; 53, 1] >> 0
242, 447

697 >> [17, 1; 41, 1] >> 0
132, 565

697 >> [17, 1; 41, 1] >> 0
319, 378

701 >> Mat([701, 1]) >> 1
135, 566

709 >> Mat([709, 1]) >> 1
96, 613

725 >> [5, 2; 29, 1] >> 0
157, 568

725 >> [5, 2; 29, 1] >> 0
307, 418

733 >> Mat([733, 1]) >> 1
353, 380

745 >> [5, 1; 149, 1] >> 0
193, 552

745 >> [5, 1; 149, 1] >> 0
342, 403

757 >> Mat([757, 1]) >> 1
87, 670

761 >> Mat([761, 1]) >> 1
39, 722

769 >> Mat([769, 1]) >> 1
62, 707

773 >> Mat([773, 1]) >> 1
317, 456

785 >> [5, 1; 157, 1] >> 0
28, 757

785 >> [5, 1; 157, 1] >> 0
342, 443

793 >> [13, 1; 61, 1] >> 0
255, 538

793 >> [13, 1; 61, 1] >> 0
294, 499

797 >> Mat([797, 1]) >> 1
215, 582

809 >> Mat([809, 1]) >> 1
318, 491

821 >> Mat([821, 1]) >> 1
295, 526

829 >> Mat([829, 1]) >> 1
246, 583

841 >> Mat([29, 2]) >> 0
41, 800

845 >> [5, 1; 13, 2] >> 0
268, 577

845 >> [5, 1; 13, 2] >> 0
408, 437

853 >> Mat([853, 1]) >> 1
333, 520

857 >> Mat([857, 1]) >> 1
207, 650

865 >> [5, 1; 173, 1] >> 0
93, 772

865 >> [5, 1; 173, 1] >> 0
253, 612

877 >> Mat([877, 1]) >> 1
151, 726

881 >> Mat([881, 1]) >> 1
387, 494

901 >> [17, 1; 53, 1] >> 0
30, 871

901 >> [17, 1; 53, 1] >> 0
242, 659

905 >> [5, 1; 181, 1] >> 0
162, 743

905 >> [5, 1; 181, 1] >> 0
343, 562

925 >> [5, 2; 37, 1] >> 0
43, 882

925 >> [5, 2; 37, 1] >> 0
68, 857

929 >> Mat([929, 1]) >> 1
324, 605

937 >> Mat([937, 1]) >> 1
196, 741

941 >> Mat([941, 1]) >> 1
97, 844

949 >> [13, 1; 73, 1] >> 0
265, 684

949 >> [13, 1; 73, 1] >> 0
411, 538

953 >> Mat([953, 1]) >> 1
442, 511

965 >> [5, 1; 193, 1] >> 0
112, 853

965 >> [5, 1; 193, 1] >> 0
467, 498

977 >> Mat([977, 1]) >> 1
252, 725

985 >> [5, 1; 197, 1] >> 0
183, 802

985 >> [5, 1; 197, 1] >> 0
408, 577

997 >> Mat([997, 1]) >> 1
161, 836

1009 >> Mat([1009, 1]) >> 1
469, 540

1013 >> Mat([1013, 1]) >> 1
45, 968

1021 >> Mat([1021, 1]) >> 1
374, 647

1025 >> [5, 2; 41, 1] >> 0
32, 993

1025 >> [5, 2; 41, 1] >> 0
132, 893

1033 >> Mat([1033, 1]) >> 1
355, 678

1037 >> [17, 1; 61, 1] >> 0
72, 965

1037 >> [17, 1; 61, 1] >> 0
438, 599

1049 >> Mat([1049, 1]) >> 1
426, 623

1061 >> Mat([1061, 1]) >> 1
103, 958

1069 >> Mat([1069, 1]) >> 1
249, 820

1073 >> [29, 1; 37, 1] >> 0
191, 882

1073 >> [29, 1; 37, 1] >> 0
302, 771

1093 >> Mat([1093, 1]) >> 1
530, 563

1097 >> Mat([1097, 1]) >> 1
341, 756

1105 >> [5, 1; 13, 1; 17, 1] >> 0
47, 1058

1105 >> [5, 1; 13, 1; 17, 1] >> 0
242, 863

1105 >> [5, 1; 13, 1; 17, 1] >> 0
268, 837

1105 >> [5, 1; 13, 1; 17, 1] >> 0
463, 642

1109 >> Mat([1109, 1]) >> 1
354, 755

1117 >> Mat([1117, 1]) >> 1
214, 903

1129 >> Mat([1129, 1]) >> 1
168, 961

1145 >> [5, 1; 229, 1] >> 0
107, 1038

1145 >> [5, 1; 229, 1] >> 0
122, 1023

1153 >> Mat([1153, 1]) >> 1
140, 1013

1157 >> [13, 1; 89, 1] >> 0
34, 1123

1157 >> [13, 1; 89, 1] >> 0
411, 746

1165 >> [5, 1; 233, 1] >> 0
322, 843

1165 >> [5, 1; 233, 1] >> 0
377, 788

1181 >> Mat([1181, 1]) >> 1
243, 938

1189 >> [29, 1; 41, 1] >> 0
278, 911

1189 >> [29, 1; 41, 1] >> 0
360, 829

1193 >> Mat([1193, 1]) >> 1
186, 1007

1201 >> Mat([1201, 1]) >> 1
49, 1152

1205 >> [5, 1; 241, 1] >> 0
177, 1028

1205 >> [5, 1; 241, 1] >> 0
418, 787

1213 >> Mat([1213, 1]) >> 1
495, 718

1217 >> Mat([1217, 1]) >> 1
78, 1139

1229 >> Mat([1229, 1]) >> 1
597, 632

1237 >> Mat([1237, 1]) >> 1
546, 691

1241 >> [17, 1; 73, 1] >> 0
319, 922

1241 >> [17, 1; 73, 1] >> 0
557, 684

1249 >> Mat([1249, 1]) >> 1
585, 664

1261 >> [13, 1; 97, 1] >> 0
216, 1045

1261 >> [13, 1; 97, 1] >> 0
463, 798

1277 >> Mat([1277, 1]) >> 1
113, 1164

1285 >> [5, 1; 257, 1] >> 0
273, 1012

1285 >> [5, 1; 257, 1] >> 0
498, 787

1289 >> Mat([1289, 1]) >> 1
479, 810

1297 >> Mat([1297, 1]) >> 1
36, 1261

1301 >> Mat([1301, 1]) >> 1
51, 1250

1313 >> [13, 1; 101, 1] >> 0
515, 798

1313 >> [13, 1; 101, 1] >> 0
616, 697

1321 >> Mat([1321, 1]) >> 1
257, 1064

1325 >> [5, 2; 53, 1] >> 0
182, 1143

1325 >> [5, 2; 53, 1] >> 0
507, 818

1345 >> [5, 1; 269, 1] >> 0
82, 1263

1345 >> [5, 1; 269, 1] >> 0
187, 1158

1361 >> Mat([1361, 1]) >> 1
614, 747

1369 >> Mat([37, 2]) >> 0
117, 1252

1373 >> Mat([1373, 1]) >> 1
668, 705

1381 >> Mat([1381, 1]) >> 1
366, 1015

1385 >> [5, 1; 277, 1] >> 0
217, 1168

1385 >> [5, 1; 277, 1] >> 0
337, 1048

1405 >> [5, 1; 281, 1] >> 0
53, 1352

1405 >> [5, 1; 281, 1] >> 0
228, 1177

1409 >> Mat([1409, 1]) >> 1
452, 957

1417 >> [13, 1; 109, 1] >> 0
294, 1123

1417 >> [13, 1; 109, 1] >> 0
512, 905

1429 >> Mat([1429, 1]) >> 1
620, 809

1433 >> Mat([1433, 1]) >> 1
542, 891

1445 >> [5, 1; 17, 2] >> 0
38, 1407

1445 >> [5, 1; 17, 2] >> 0
327, 1118

1453 >> Mat([1453, 1]) >> 1
497, 956

1465 >> [5, 1; 293, 1] >> 0
138, 1327

1465 >> [5, 1; 293, 1] >> 0
448, 1017

1469 >> [13, 1; 113, 1] >> 0
437, 1032

1469 >> [13, 1; 113, 1] >> 0
580, 889

1481 >> Mat([1481, 1]) >> 1
465, 1016

1489 >> Mat([1489, 1]) >> 1
225, 1264

1493 >> Mat([1493, 1]) >> 1
432, 1061

1513 >> [17, 1; 89, 1] >> 0
55, 1458

1513 >> [17, 1; 89, 1] >> 0
123, 1390

1517 >> [37, 1; 41, 1] >> 0
401, 1116

1517 >> [37, 1; 41, 1] >> 0
524, 993

1525 >> [5, 2; 61, 1] >> 0
682, 843

1525 >> [5, 2; 61, 1] >> 0
743, 782

1537 >> [29, 1; 53, 1] >> 0
394, 1143

1537 >> [29, 1; 53, 1] >> 0
447, 1090

1549 >> Mat([1549, 1]) >> 1
88, 1461

1553 >> Mat([1553, 1]) >> 1
339, 1214

1565 >> [5, 1; 313, 1] >> 0
288, 1277

1565 >> [5, 1; 313, 1] >> 0
338, 1227

1585 >> [5, 1; 317, 1] >> 0
203, 1382

1585 >> [5, 1; 317, 1] >> 0
748, 837

1597 >> Mat([1597, 1]) >> 1
610, 987

1601 >> Mat([1601, 1]) >> 1
40, 1561

1609 >> Mat([1609, 1]) >> 1
523, 1086

1613 >> Mat([1613, 1]) >> 1
127, 1486

1621 >> Mat([1621, 1]) >> 1
166, 1455

1625 >> [5, 3; 13, 1] >> 0
57, 1568

1625 >> [5, 3; 13, 1] >> 0
307, 1318

1637 >> Mat([1637, 1]) >> 1
316, 1321

1649 >> [17, 1; 97, 1] >> 0
463, 1186

1649 >> [17, 1; 97, 1] >> 0
701, 948

1657 >> Mat([1657, 1]) >> 1
783, 874

1669 >> Mat([1669, 1]) >> 1
220, 1449

1681 >> Mat([41, 2]) >> 0
378, 1303

1685 >> [5, 1; 337, 1] >> 0
148, 1537

1685 >> [5, 1; 337, 1] >> 0
822, 863

1693 >> Mat([1693, 1]) >> 1
92, 1601

1697 >> Mat([1697, 1]) >> 1
414, 1283

1709 >> Mat([1709, 1]) >> 1
390, 1319

1717 >> [17, 1; 101, 1] >> 0
293, 1424

1717 >> [17, 1; 101, 1] >> 0
616, 1101

1721 >> Mat([1721, 1]) >> 1
473, 1248

1733 >> Mat([1733, 1]) >> 1
410, 1323

1741 >> Mat([1741, 1]) >> 1
59, 1682

1745 >> [5, 1; 349, 1] >> 0
213, 1532

1745 >> [5, 1; 349, 1] >> 0
562, 1183

1753 >> Mat([1753, 1]) >> 1
713, 1040

1765 >> [5, 1; 353, 1] >> 0
42, 1723

1765 >> [5, 1; 353, 1] >> 0
748, 1017

1769 >> [29, 1; 61, 1] >> 0
133, 1636

1769 >> [29, 1; 61, 1] >> 0
621, 1148

1777 >> Mat([1777, 1]) >> 1
775, 1002

1781 >> [13, 1; 137, 1] >> 0
174, 1607

1781 >> [13, 1; 137, 1] >> 0
785, 996

1789 >> Mat([1789, 1]) >> 1
724, 1065

1801 >> Mat([1801, 1]) >> 1
824, 977

1825 >> [5, 2; 73, 1] >> 0
557, 1268

1825 >> [5, 2; 73, 1] >> 0
757, 1068

1853 >> [17, 1; 109, 1] >> 0
251, 1602

1853 >> [17, 1; 109, 1] >> 0
905, 948

1861 >> Mat([1861, 1]) >> 1
61, 1800

1865 >> [5, 1; 373, 1] >> 0
477, 1388

1865 >> [5, 1; 373, 1] >> 0
642, 1223

1873 >> Mat([1873, 1]) >> 1
737, 1136

1877 >> Mat([1877, 1]) >> 1
137, 1740

1885 >> [5, 1; 13, 1; 29, 1] >> 0
278, 1607

1885 >> [5, 1; 13, 1; 29, 1] >> 0
307, 1578

1885 >> [5, 1; 13, 1; 29, 1] >> 0
447, 1438

1885 >> [5, 1; 13, 1; 29, 1] >> 0
853, 1032

1889 >> Mat([1889, 1]) >> 1
331, 1558

1901 >> Mat([1901, 1]) >> 1
218, 1683

1913 >> Mat([1913, 1]) >> 1
712, 1201

1921 >> [17, 1; 113, 1] >> 0
98, 1823

1921 >> [17, 1; 113, 1] >> 0
693, 1228

1933 >> Mat([1933, 1]) >> 1
598, 1335

1937 >> [13, 1; 149, 1] >> 0
44, 1893

1937 >> [13, 1; 149, 1] >> 0
850, 1087

1945 >> [5, 1; 389, 1] >> 0
663, 1282

1945 >> [5, 1; 389, 1] >> 0
893, 1052

1949 >> Mat([1949, 1]) >> 1
589, 1360

1961 >> [37, 1; 53, 1] >> 0
401, 1560

1961 >> [37, 1; 53, 1] >> 0
931, 1030

1973 >> Mat([1973, 1]) >> 1
259, 1714

1985 >> [5, 1; 397, 1] >> 0
63, 1922

1985 >> [5, 1; 397, 1] >> 0
857, 1128

1993 >> Mat([1993, 1]) >> 1
834, 1159

1997 >> Mat([1997, 1]) >> 1
412, 1585

2005 >> [5, 1; 401, 1] >> 0
782, 1223

2005 >> [5, 1; 401, 1] >> 0
822, 1183

2017 >> Mat([2017, 1]) >> 1
229, 1788

2029 >> Mat([2029, 1]) >> 1
992, 1037

2041 >> [13, 1; 157, 1] >> 0
499, 1542

2041 >> [13, 1; 157, 1] >> 0
970, 1071

2045 >> [5, 1; 409, 1] >> 0
143, 1902

2045 >> [5, 1; 409, 1] >> 0
552, 1493

2053 >> Mat([2053, 1]) >> 1
244, 1809

2069 >> Mat([2069, 1]) >> 1
164, 1905

2081 >> Mat([2081, 1]) >> 1
102, 1979

2089 >> Mat([2089, 1]) >> 1
789, 1300

2105 >> [5, 1; 421, 1] >> 0
392, 1713

2105 >> [5, 1; 421, 1] >> 0
813, 1292

2113 >> Mat([2113, 1]) >> 1
65, 2048

2117 >> [29, 1; 73, 1] >> 0
46, 2071

2117 >> [29, 1; 73, 1] >> 0
684, 1433

2125 >> [5, 3; 17, 1] >> 0
557, 1568

2125 >> [5, 3; 17, 1] >> 0
693, 1432

2129 >> Mat([2129, 1]) >> 1
372, 1757

2137 >> Mat([2137, 1]) >> 1
296, 1841

2141 >> Mat([2141, 1]) >> 1
419, 1722

2153 >> Mat([2153, 1]) >> 1
232, 1921

2161 >> Mat([2161, 1]) >> 1
147, 2014

2165 >> [5, 1; 433, 1] >> 0
612, 1553

2165 >> [5, 1; 433, 1] >> 0
687, 1478

2173 >> [41, 1; 53, 1] >> 0
401, 1772

2173 >> [41, 1; 53, 1] >> 0
606, 1567

2197 >> Mat([13, 3]) >> 0
239, 1958

2213 >> Mat([2213, 1]) >> 1
1083, 1130

2221 >> Mat([2221, 1]) >> 1
790, 1431

2225 >> [5, 2; 89, 1] >> 0
568, 1657

2225 >> [5, 2; 89, 1] >> 0
657, 1568

2237 >> Mat([2237, 1]) >> 1
1021, 1216

2245 >> [5, 1; 449, 1] >> 0
67, 2178

2245 >> [5, 1; 449, 1] >> 0
382, 1863

2249 >> [13, 1; 173, 1] >> 0
772, 1477

2249 >> [13, 1; 173, 1] >> 0
785, 1464

2257 >> [37, 1; 61, 1] >> 0
438, 1819

2257 >> [37, 1; 61, 1] >> 0
660, 1597

2269 >> Mat([2269, 1]) >> 1
982, 1287

2273 >> Mat([2273, 1]) >> 1
290, 1983

2281 >> Mat([2281, 1]) >> 1
710, 1571

2285 >> [5, 1; 457, 1] >> 0
348, 1937

2285 >> [5, 1; 457, 1] >> 0
1023, 1262

2293 >> Mat([2293, 1]) >> 1
600, 1693

2297 >> Mat([2297, 1]) >> 1
365, 1932

2305 >> [5, 1; 461, 1] >> 0
48, 2257

2305 >> [5, 1; 461, 1] >> 0
413, 1892

2309 >> Mat([2309, 1]) >> 1
688, 1621

2329 >> [17, 1; 137, 1] >> 0
174, 2155

2329 >> [17, 1; 137, 1] >> 0
922, 1407

2333 >> Mat([2333, 1]) >> 1
108, 2225

2341 >> Mat([2341, 1]) >> 1
153, 2188

2353 >> [13, 1; 181, 1] >> 0
200, 2153

2353 >> [13, 1; 181, 1] >> 0
343, 2010

2357 >> Mat([2357, 1]) >> 1
633, 1724

2377 >> Mat([2377, 1]) >> 1
1134, 1243

2381 >> Mat([2381, 1]) >> 1
69, 2312

2389 >> Mat([2389, 1]) >> 1
285, 2104

2393 >> Mat([2393, 1]) >> 1
971, 1422

2405 >> [5, 1; 13, 1; 37, 1] >> 0
512, 1893

2405 >> [5, 1; 13, 1; 37, 1] >> 0
697, 1708

2405 >> [5, 1; 13, 1; 37, 1] >> 0
993, 1412

2405 >> [5, 1; 13, 1; 37, 1] >> 0
1178, 1227

2417 >> Mat([2417, 1]) >> 1
592, 1825

2425 >> [5, 2; 97, 1] >> 0
507, 1918

2425 >> [5, 2; 97, 1] >> 0
657, 1768

2437 >> Mat([2437, 1]) >> 1
398, 2039

2441 >> Mat([2441, 1]) >> 1
672, 1769

2465 >> [5, 1; 17, 1; 29, 1] >> 0
157, 2308

2465 >> [5, 1; 17, 1; 29, 1] >> 0
302, 2163

2465 >> [5, 1; 17, 1; 29, 1] >> 0
1143, 1322

2465 >> [5, 1; 17, 1; 29, 1] >> 0
1177, 1288

2473 >> Mat([2473, 1]) >> 1
567, 1906

2477 >> Mat([2477, 1]) >> 1
915, 1562

2501 >> [41, 1; 61, 1] >> 0
50, 2451

2501 >> [41, 1; 61, 1] >> 0
255, 2246

2509 >> [13, 1; 193, 1] >> 0
112, 2397

2509 >> [13, 1; 193, 1] >> 0
853, 1656

2521 >> Mat([2521, 1]) >> 1
71, 2450

2525 >> [5, 2; 101, 1] >> 0
293, 2232

2525 >> [5, 2; 101, 1] >> 0
818, 1707

2533 >> [17, 1; 149, 1] >> 0
701, 1832

2533 >> [17, 1; 149, 1] >> 0
999, 1534

2545 >> [5, 1; 509, 1] >> 0
208, 2337

2545 >> [5, 1; 509, 1] >> 0
717, 1828

2549 >> Mat([2549, 1]) >> 1
357, 2192

2557 >> Mat([2557, 1]) >> 1
611, 1946

2561 >> [13, 1; 197, 1] >> 0
408, 2153

2561 >> [13, 1; 197, 1] >> 0
577, 1984

2581 >> [29, 1; 89, 1] >> 0
568, 2013

2581 >> [29, 1; 89, 1] >> 0
945, 1636

2593 >> Mat([2593, 1]) >> 1
918, 1675

2605 >> [5, 1; 521, 1] >> 0
807, 1798

2605 >> [5, 1; 521, 1] >> 0
1277, 1328

2609 >> Mat([2609, 1]) >> 1
389, 2220

2617 >> Mat([2617, 1]) >> 1
667, 1950

2621 >> Mat([2621, 1]) >> 1
472, 2149

2633 >> Mat([2633, 1]) >> 1
1224, 1409

2657 >> Mat([2657, 1]) >> 1
163, 2494

2665 >> [5, 1; 13, 1; 41, 1] >> 0
73, 2592

2665 >> [5, 1; 13, 1; 41, 1] >> 0
278, 2387

2665 >> [5, 1; 13, 1; 41, 1] >> 0
788, 1877

2665 >> [5, 1; 13, 1; 41, 1] >> 0
993, 1672

2669 >> [17, 1; 157, 1] >> 0
914, 1755

2669 >> [17, 1; 157, 1] >> 0
1228, 1441

2677 >> Mat([2677, 1]) >> 1
550, 2127

2689 >> Mat([2689, 1]) >> 1
1142, 1547

2693 >> Mat([2693, 1]) >> 1
859, 1834

2701 >> [37, 1; 73, 1] >> 0
265, 2436

2701 >> [37, 1; 73, 1] >> 0
1141, 1560

2705 >> [5, 1; 541, 1] >> 0
52, 2653

2705 >> [5, 1; 541, 1] >> 0
593, 2112

2713 >> Mat([2713, 1]) >> 1
887, 1826

2725 >> [5, 2; 109, 1] >> 0
1057, 1668

2725 >> [5, 2; 109, 1] >> 0
1232, 1493

2729 >> Mat([2729, 1]) >> 1
1102, 1627

2741 >> Mat([2741, 1]) >> 1
656, 2085

2749 >> Mat([2749, 1]) >> 1
640, 2109

2753 >> Mat([2753, 1]) >> 1
794, 1959

2777 >> Mat([2777, 1]) >> 1
190, 2587

2785 >> [5, 1; 557, 1] >> 0
118, 2667

2785 >> [5, 1; 557, 1] >> 0
1232, 1553

2789 >> Mat([2789, 1]) >> 1
167, 2622

2797 >> Mat([2797, 1]) >> 1
603, 2194

2801 >> Mat([2801, 1]) >> 1
1258, 1543

2809 >> Mat([53, 2]) >> 0
500, 2309

2813 >> [29, 1; 97, 1] >> 0
75, 2738

2813 >> [29, 1; 97, 1] >> 0
1380, 1433

2825 >> [5, 2; 113, 1] >> 0
693, 2132

2825 >> [5, 2; 113, 1] >> 0
1032, 1793

2833 >> Mat([2833, 1]) >> 1
1357, 1476

2837 >> Mat([2837, 1]) >> 1
416, 2421

2845 >> [5, 1; 569, 1] >> 0
483, 2362

2845 >> [5, 1; 569, 1] >> 0
1052, 1793

2857 >> Mat([2857, 1]) >> 1
896, 1961

2861 >> Mat([2861, 1]) >> 1
1202, 1659

2873 >> [13, 2; 17, 1] >> 0
268, 2605

2873 >> [13, 2; 17, 1] >> 0
1084, 1789

2885 >> [5, 1; 577, 1] >> 0
553, 2332

2885 >> [5, 1; 577, 1] >> 0
1178, 1707

2897 >> Mat([2897, 1]) >> 1
1120, 1777

2909 >> Mat([2909, 1]) >> 1
878, 2031

2917 >> Mat([2917, 1]) >> 1
54, 2863

2929 >> [29, 1; 101, 1] >> 0
394, 2535

2929 >> [29, 1; 101, 1] >> 0
1404, 1525

2941 >> [17, 1; 173, 1] >> 0
599, 2342

2941 >> [17, 1; 173, 1] >> 0
1118, 1823

2953 >> Mat([2953, 1]) >> 1
1226, 1727

2957 >> Mat([2957, 1]) >> 1
1222, 1735

2965 >> [5, 1; 593, 1] >> 0
77, 2888

2965 >> [5, 1; 593, 1] >> 0
1263, 1702

2969 >> Mat([2969, 1]) >> 1
964, 2005

2977 >> [13, 1; 229, 1] >> 0
122, 2855

2977 >> [13, 1; 229, 1] >> 0
580, 2397

2993 >> [41, 1; 73, 1] >> 0
173, 2820

2993 >> [41, 1; 73, 1] >> 0
319, 2674

3001 >> Mat([3001, 1]) >> 1
1353, 1648

3005 >> [5, 1; 601, 1] >> 0
1077, 1928

3005 >> [5, 1; 601, 1] >> 0
1327, 1678

3029 >> [13, 1; 233, 1] >> 0
788, 2241

3029 >> [13, 1; 233, 1] >> 0
1487, 1542

3037 >> Mat([3037, 1]) >> 1
281, 2756

3041 >> Mat([3041, 1]) >> 1
774, 2267

3049 >> Mat([3049, 1]) >> 1
475, 2574

3061 >> Mat([3061, 1]) >> 1
501, 2560

3065 >> [5, 1; 613, 1] >> 0
578, 2487

3065 >> [5, 1; 613, 1] >> 0
648, 2417

3077 >> [17, 1; 181, 1] >> 0
200, 2877

3077 >> [17, 1; 181, 1] >> 0
1067, 2010

3085 >> [5, 1; 617, 1] >> 0
423, 2662

3085 >> [5, 1; 617, 1] >> 0
1428, 1657

3089 >> Mat([3089, 1]) >> 1
393, 2696

3109 >> Mat([3109, 1]) >> 1
727, 2382

3121 >> Mat([3121, 1]) >> 1
79, 3042

3125 >> Mat([5, 5]) >> 0
1068, 2057

3133 >> [13, 1; 241, 1] >> 0
177, 2956

3133 >> [13, 1; 241, 1] >> 0
1269, 1864

3137 >> Mat([3137, 1]) >> 1
56, 3081

3145 >> [5, 1; 17, 1; 37, 1] >> 0
302, 2843

3145 >> [5, 1; 17, 1; 37, 1] >> 0
327, 2818

3145 >> [5, 1; 17, 1; 37, 1] >> 0
438, 2707

3145 >> [5, 1; 17, 1; 37, 1] >> 0
1067, 2078

3161 >> [29, 1; 109, 1] >> 0
360, 2801

3161 >> [29, 1; 109, 1] >> 0
621, 2540

3169 >> Mat([3169, 1]) >> 1
1325, 1844

3181 >> Mat([3181, 1]) >> 1
282, 2899

3205 >> [5, 1; 641, 1] >> 0
487, 2718

3205 >> [5, 1; 641, 1] >> 0
1128, 2077

3209 >> Mat([3209, 1]) >> 1
484, 2725

3217 >> Mat([3217, 1]) >> 1
1436, 1781

3221 >> Mat([3221, 1]) >> 1
234, 2987

3229 >> Mat([3229, 1]) >> 1
839, 2390

3233 >> [53, 1; 61, 1] >> 0
560, 2673

3233 >> [53, 1; 61, 1] >> 0
1514, 1719

3253 >> Mat([3253, 1]) >> 1
1598, 1655

3257 >> Mat([3257, 1]) >> 1
291, 2966

3265 >> [5, 1; 653, 1] >> 0
802, 2463

3265 >> [5, 1; 653, 1] >> 0
1157, 2108

3277 >> [29, 1; 113, 1] >> 0
128, 3149

3277 >> [29, 1; 113, 1] >> 0
1032, 2245

3281 >> [17, 1; 193, 1] >> 0
81, 3200

3281 >> [17, 1; 193, 1] >> 0
1432, 1849

3293 >> [37, 1; 89, 1] >> 0
746, 2547

3293 >> [37, 1; 89, 1] >> 0
1301, 1992

3301 >> Mat([3301, 1]) >> 1
1212, 2089

3305 >> [5, 1; 661, 1] >> 0
767, 2538

3305 >> [5, 1; 661, 1] >> 0
1428, 1877

3313 >> Mat([3313, 1]) >> 1
407, 2906

3329 >> Mat([3329, 1]) >> 1
1600, 1729

3341 >> [13, 1; 257, 1] >> 0
1269, 2072

3341 >> [13, 1; 257, 1] >> 0
1526, 1815

3349 >> [17, 1; 197, 1] >> 0
183, 3166

3349 >> [17, 1; 197, 1] >> 0
999, 2350

3361 >> Mat([3361, 1]) >> 1
900, 2461

3365 >> [5, 1; 673, 1] >> 0
58, 3307

3365 >> [5, 1; 673, 1] >> 0
1288, 2077

3373 >> Mat([3373, 1]) >> 1
1105, 2268

3385 >> [5, 1; 677, 1] >> 0
703, 2682

3385 >> [5, 1; 677, 1] >> 0
1328, 2057

3389 >> Mat([3389, 1]) >> 1
1344, 2045

3413 >> Mat([3413, 1]) >> 1
1471, 1942

3425 >> [5, 2; 137, 1] >> 0
1407, 2018

3425 >> [5, 2; 137, 1] >> 0
1607, 1818

3433 >> Mat([3433, 1]) >> 1
1651, 1782

3445 >> [5, 1; 13, 1; 53, 1] >> 0
83, 3362

3445 >> [5, 1; 13, 1; 53, 1] >> 0
242, 3203

3445 >> [5, 1; 13, 1; 53, 1] >> 0
447, 2998

3445 >> [5, 1; 13, 1; 53, 1] >> 0
772, 2673

3449 >> Mat([3449, 1]) >> 1
1122, 2327

3457 >> Mat([3457, 1]) >> 1
708, 2749

3461 >> Mat([3461, 1]) >> 1
1453, 2008

3469 >> Mat([3469, 1]) >> 1
1003, 2466

3485 >> [5, 1; 17, 1; 41, 1] >> 0
132, 3353

3485 >> [5, 1; 17, 1; 41, 1] >> 0
378, 3107

3485 >> [5, 1; 17, 1; 41, 1] >> 0
1262, 2223

3485 >> [5, 1; 17, 1; 41, 1] >> 0
1713, 1772

3497 >> [13, 1; 269, 1] >> 0
187, 3310

3497 >> [13, 1; 269, 1] >> 0
889, 2608

3505 >> [5, 1; 701, 1] >> 0
1267, 2238

3505 >> [5, 1; 701, 1] >> 0
1537, 1968

3517 >> Mat([3517, 1]) >> 1
596, 2921

3529 >> Mat([3529, 1]) >> 1
808, 2721

3533 >> Mat([3533, 1]) >> 1
548, 2985

3541 >> Mat([3541, 1]) >> 1
852, 2689

3545 >> [5, 1; 709, 1] >> 0
613, 2932

3545 >> [5, 1; 709, 1] >> 0
1322, 2223

3557 >> Mat([3557, 1]) >> 1
943, 2614

3581 >> Mat([3581, 1]) >> 1
364, 3217

3589 >> [37, 1; 97, 1] >> 0
216, 3373

3589 >> [37, 1; 97, 1] >> 0
1671, 1918

3593 >> Mat([3593, 1]) >> 1
1153, 2440

3601 >> [13, 1; 277, 1] >> 0
60, 3541

3601 >> [13, 1; 277, 1] >> 0
1048, 2553

3613 >> Mat([3613, 1]) >> 1
85, 3528

3617 >> Mat([3617, 1]) >> 1
1234, 2383

3625 >> [5, 3; 29, 1] >> 0
307, 3318

3625 >> [5, 3; 29, 1] >> 0
568, 3057

3637 >> Mat([3637, 1]) >> 1
1027, 2610

3649 >> [41, 1; 89, 1] >> 0
1034, 2615

3649 >> [41, 1; 89, 1] >> 0
1280, 2369

3653 >> [13, 1; 281, 1] >> 0
1071, 2582

3653 >> [13, 1; 281, 1] >> 0
1633, 2020

3665 >> [5, 1; 733, 1] >> 0
353, 3312

3665 >> [5, 1; 733, 1] >> 0
1113, 2552

3673 >> Mat([3673, 1]) >> 1
994, 2679

3677 >> Mat([3677, 1]) >> 1
1309, 2368

3697 >> Mat([3697, 1]) >> 1
1131, 2566

3701 >> Mat([3701, 1]) >> 1
1279, 2422

3709 >> Mat([3709, 1]) >> 1
1609, 2100

3721 >> Mat([61, 2]) >> 0
682, 3039

3725 >> [5, 2; 149, 1] >> 0
193, 3532

3725 >> [5, 2; 149, 1] >> 0
1832, 1893

3733 >> Mat([3733, 1]) >> 1
851, 2882

3737 >> [37, 1; 101, 1] >> 0
697, 3040

3737 >> [37, 1; 101, 1] >> 0
919, 2818

3757 >> [13, 1; 17, 2] >> 0
616, 3141

3757 >> [13, 1; 17, 2] >> 0
905, 2852

3761 >> Mat([3761, 1]) >> 1
604, 3157

3769 >> Mat([3769, 1]) >> 1
1445, 2324

3785 >> [5, 1; 757, 1] >> 0
87, 3698

3785 >> [5, 1; 757, 1] >> 0
1427, 2358

3793 >> Mat([3793, 1]) >> 1
803, 2990

3797 >> Mat([3797, 1]) >> 1
742, 3055

3805 >> [5, 1; 761, 1] >> 0
722, 3083

3805 >> [5, 1; 761, 1] >> 0
1483, 2322

3809 >> [13, 1; 293, 1] >> 0
138, 3671

3809 >> [13, 1; 293, 1] >> 0
1620, 2189

3821 >> Mat([3821, 1]) >> 1
376, 3445

3833 >> Mat([3833, 1]) >> 1
361, 3472

3845 >> [5, 1; 769, 1] >> 0
62, 3783

3845 >> [5, 1; 769, 1] >> 0
707, 3138

3853 >> Mat([3853, 1]) >> 1
1305, 2548

3865 >> [5, 1; 773, 1] >> 0
317, 3548

3865 >> [5, 1; 773, 1] >> 0
1863, 2002

3869 >> [53, 1; 73, 1] >> 0
1560, 2309

3869 >> [53, 1; 73, 1] >> 0
1779, 2090

3877 >> Mat([3877, 1]) >> 1
502, 3375

3881 >> Mat([3881, 1]) >> 1
197, 3684

3889 >> Mat([3889, 1]) >> 1
454, 3435

3893 >> [17, 1; 229, 1] >> 0
336, 3557

3893 >> [17, 1; 229, 1] >> 0
565, 3328

3917 >> Mat([3917, 1]) >> 1
835, 3082

3925 >> [5, 2; 157, 1] >> 0
443, 3482

3925 >> [5, 2; 157, 1] >> 0
757, 3168

3929 >> Mat([3929, 1]) >> 1
226, 3703

3961 >> [17, 1; 233, 1] >> 0
89, 3872

3961 >> [17, 1; 233, 1] >> 0
1254, 2707

3965 >> [5, 1; 13, 1; 61, 1] >> 0
538, 3427

3965 >> [5, 1; 13, 1; 61, 1] >> 0
1048, 2917

3965 >> [5, 1; 13, 1; 61, 1] >> 0
1087, 2878

3965 >> [5, 1; 13, 1; 61, 1] >> 0
1292, 2673

3973 >> [29, 1; 137, 1] >> 0
1607, 2366

3973 >> [29, 1; 137, 1] >> 0
1955, 2018

3977 >> [41, 1; 97, 1] >> 0
1239, 2738

3977 >> [41, 1; 97, 1] >> 0
1918, 2059

3985 >> [5, 1; 797, 1] >> 0
582, 3403

3985 >> [5, 1; 797, 1] >> 0
1012, 2973

3989 >> Mat([3989, 1]) >> 1
481, 3508

4001 >> Mat([4001, 1]) >> 1
899, 3102

4013 >> Mat([4013, 1]) >> 1
1230, 2783

4021 >> Mat([4021, 1]) >> 1
723, 3298

4033 >> [37, 1; 109, 1] >> 0
142, 3891

4033 >> [37, 1; 109, 1] >> 0
512, 3521

4045 >> [5, 1; 809, 1] >> 0
318, 3727

4045 >> [5, 1; 809, 1] >> 0
1127, 2918

4049 >> Mat([4049, 1]) >> 1
884, 3165

4057 >> Mat([4057, 1]) >> 1
1857, 2200

4069 >> [13, 1; 313, 1] >> 0
1227, 2842

4069 >> [13, 1; 313, 1] >> 0
1903, 2166

4073 >> Mat([4073, 1]) >> 1
549, 3524

4093 >> Mat([4093, 1]) >> 1
1059, 3034

4097 >> [17, 1; 241, 1] >> 0
64, 4033

4097 >> [17, 1; 241, 1] >> 0
659, 3438

4105 >> [5, 1; 821, 1] >> 0
1347, 2758

4105 >> [5, 1; 821, 1] >> 0
1937, 2168

4121 >> [13, 1; 317, 1] >> 0
203, 3918

4121 >> [13, 1; 317, 1] >> 0
837, 3284

4129 >> Mat([4129, 1]) >> 1
895, 3234

4133 >> Mat([4133, 1]) >> 1
733, 3400

4141 >> [41, 1; 101, 1] >> 0
91, 4050

4141 >> [41, 1; 101, 1] >> 0
1303, 2838

4145 >> [5, 1; 829, 1] >> 0
583, 3562

4145 >> [5, 1; 829, 1] >> 0
1412, 2733

4153 >> Mat([4153, 1]) >> 1
1643, 2510

4157 >> Mat([4157, 1]) >> 1
1761, 2396

4177 >> Mat([4177, 1]) >> 1
457, 3720

4181 >> [37, 1; 113, 1] >> 0
919, 3262

4181 >> [37, 1; 113, 1] >> 0
1597, 2584

4201 >> Mat([4201, 1]) >> 1
1154, 3047

4205 >> [5, 1; 29, 2] >> 0
882, 3323

4205 >> [5, 1; 29, 2] >> 0
1723, 2482

4217 >> Mat([4217, 1]) >> 1
1911, 2306

4225 >> [5, 2; 13, 2] >> 0
268, 3957

4225 >> [5, 2; 13, 2] >> 0
1282, 2943

4229 >> Mat([4229, 1]) >> 1
2082, 2147

4241 >> Mat([4241, 1]) >> 1
1044, 3197

4253 >> Mat([4253, 1]) >> 1
561, 3692

4261 >> Mat([4261, 1]) >> 1
721, 3540

4265 >> [5, 1; 853, 1] >> 0
333, 3932

4265 >> [5, 1; 853, 1] >> 0
1373, 2892

4273 >> Mat([4273, 1]) >> 1
1200, 3073

4285 >> [5, 1; 857, 1] >> 0
207, 4078

4285 >> [5, 1; 857, 1] >> 0
1507, 2778

4289 >> Mat([4289, 1]) >> 1
528, 3761

4297 >> Mat([4297, 1]) >> 1
1972, 2325

4321 >> [29, 1; 149, 1] >> 0
1148, 3173

4321 >> [29, 1; 149, 1] >> 0
2042, 2279

4325 >> [5, 2; 173, 1] >> 0
93, 4232

4325 >> [5, 2; 173, 1] >> 0
1118, 3207

4337 >> Mat([4337, 1]) >> 1
886, 3451

4349 >> Mat([4349, 1]) >> 1
608, 3741

4357 >> Mat([4357, 1]) >> 1
66, 4291

4369 >> [17, 1; 257, 1] >> 0
1526, 2843

4369 >> [17, 1; 257, 1] >> 0
1815, 2554

4373 >> Mat([4373, 1]) >> 1
1904, 2469

4381 >> [13, 1; 337, 1] >> 0
148, 4233

4381 >> [13, 1; 337, 1] >> 0
863, 3518

4385 >> [5, 1; 877, 1] >> 0
1028, 3357

4385 >> [5, 1; 877, 1] >> 0
1603, 2782

4397 >> Mat([4397, 1]) >> 1
505, 3892

4405 >> [5, 1; 881, 1] >> 0
387, 4018

4405 >> [5, 1; 881, 1] >> 0
1268, 3137

4409 >> Mat([4409, 1]) >> 1
332, 4077

4421 >> Mat([4421, 1]) >> 1
952, 3469

4441 >> Mat([4441, 1]) >> 1
2146, 2295

4453 >> [61, 1; 73, 1] >> 0
538, 3915

4453 >> [61, 1; 73, 1] >> 0
1414, 3039

4457 >> Mat([4457, 1]) >> 1
1880, 2577

4469 >> [41, 1; 109, 1] >> 0
360, 4109

4469 >> [41, 1; 109, 1] >> 0
1057, 3412

4481 >> Mat([4481, 1]) >> 1
276, 4205

4493 >> Mat([4493, 1]) >> 1
2213, 2280

4505 >> [5, 1; 17, 1; 53, 1] >> 0
242, 4263

4505 >> [5, 1; 17, 1; 53, 1] >> 0
1143, 3362

4505 >> [5, 1; 17, 1; 53, 1] >> 0
1772, 2733

4505 >> [5, 1; 17, 1; 53, 1] >> 0
1832, 2673

4513 >> Mat([4513, 1]) >> 1
95, 4418

4517 >> Mat([4517, 1]) >> 1
1474, 3043

4525 >> [5, 2; 181, 1] >> 0
343, 4182

4525 >> [5, 2; 181, 1] >> 0
743, 3782

4537 >> [13, 1; 349, 1] >> 0
213, 4324

4537 >> [13, 1; 349, 1] >> 0
1958, 2579

4549 >> Mat([4549, 1]) >> 1
1260, 3289

4553 >> [29, 1; 157, 1] >> 0
1699, 2854

4553 >> [29, 1; 157, 1] >> 0
2013, 2540

4561 >> Mat([4561, 1]) >> 1
2205, 2356

4573 >> [17, 1; 269, 1] >> 0
1696, 2877

4573 >> [17, 1; 269, 1] >> 0
2070, 2503

4589 >> [13, 1; 353, 1] >> 0
395, 4194

4589 >> [13, 1; 353, 1] >> 0
1370, 3219

4597 >> Mat([4597, 1]) >> 1
2129, 2468

4621 >> Mat([4621, 1]) >> 1
152, 4469

4625 >> [5, 3; 37, 1] >> 0
68, 4557

4625 >> [5, 3; 37, 1] >> 0
1807, 2818

4633 >> [41, 1; 113, 1] >> 0
1567, 3066

4633 >> [41, 1; 113, 1] >> 0
1936, 2697

4637 >> Mat([4637, 1]) >> 1
2044, 2593

4645 >> [5, 1; 929, 1] >> 0
1253, 3392

4645 >> [5, 1; 929, 1] >> 0
2182, 2463

4649 >> Mat([4649, 1]) >> 1
1846, 2803

4657 >> Mat([4657, 1]) >> 1
1912, 2745

4673 >> Mat([4673, 1]) >> 1
1993, 2680

4685 >> [5, 1; 937, 1] >> 0
1133, 3552

4685 >> [5, 1; 937, 1] >> 0
1678, 3007

4705 >> [5, 1; 941, 1] >> 0
97, 4608

4705 >> [5, 1; 941, 1] >> 0
1038, 3667

4709 >> [17, 1; 277, 1] >> 0
217, 4492

4709 >> [17, 1; 277, 1] >> 0
1602, 3107

4717 >> [53, 1; 89, 1] >> 0
500, 4217

4717 >> [53, 1; 89, 1] >> 0
924, 3793

4721 >> Mat([4721, 1]) >> 1
1697, 3024

4729 >> Mat([4729, 1]) >> 1
1365, 3364

4733 >> Mat([4733, 1]) >> 1
897, 3836

4745 >> [5, 1; 13, 1; 73, 1] >> 0
538, 4207

4745 >> [5, 1; 13, 1; 73, 1] >> 0
1487, 3258

4745 >> [5, 1; 13, 1; 73, 1] >> 0
1633, 3112

4745 >> [5, 1; 13, 1; 73, 1] >> 0
2163, 2582

4765 >> [5, 1; 953, 1] >> 0
442, 4323

4765 >> [5, 1; 953, 1] >> 0
2348, 2417

4777 >> [17, 1; 281, 1] >> 0
1177, 3600

4777 >> [17, 1; 281, 1] >> 0
1458, 3319

4789 >> Mat([4789, 1]) >> 1
1481, 3308

4793 >> Mat([4793, 1]) >> 1
1480, 3313

4801 >> Mat([4801, 1]) >> 1
1403, 3398

4813 >> Mat([4813, 1]) >> 1
1868, 2945

4817 >> Mat([4817, 1]) >> 1
1291, 3526

4825 >> [5, 2; 193, 1] >> 0
1432, 3393

4825 >> [5, 2; 193, 1] >> 0
1818, 3007

4849 >> [13, 1; 373, 1] >> 0
642, 4207

4849 >> [13, 1; 373, 1] >> 0
850, 3999

4861 >> Mat([4861, 1]) >> 1
493, 4368

4877 >> Mat([4877, 1]) >> 1
719, 4158

4885 >> [5, 1; 977, 1] >> 0
252, 4633

4885 >> [5, 1; 977, 1] >> 0
1702, 3183

4889 >> Mat([4889, 1]) >> 1
730, 4159

4901 >> [13, 2; 29, 1] >> 0
70, 4831

4901 >> [13, 2; 29, 1] >> 0
99, 4802

4909 >> Mat([4909, 1]) >> 1
1613, 3296

4913 >> Mat([17, 3]) >> 0
1985, 2928

4925 >> [5, 2; 197, 1] >> 0
1168, 3757

4925 >> [5, 2; 197, 1] >> 0
1393, 3532

4933 >> Mat([4933, 1]) >> 1
1194, 3739

4937 >> Mat([4937, 1]) >> 1
849, 4088

4957 >> Mat([4957, 1]) >> 1
359, 4598

4969 >> Mat([4969, 1]) >> 1
1076, 3893

4973 >> Mat([4973, 1]) >> 1
223, 4750

4981 >> [17, 1; 293, 1] >> 0
2189, 2792

4981 >> [17, 1; 293, 1] >> 0
2206, 2775

4985 >> [5, 1; 997, 1] >> 0
1158, 3827

4985 >> [5, 1; 997, 1] >> 0
1833, 3152

4993 >> Mat([4993, 1]) >> 1
158, 4835

5009 >> Mat([5009, 1]) >> 1
539, 4470

5017 >> [29, 1; 173, 1] >> 0
945, 4072

5017 >> [29, 1; 173, 1] >> 0
1810, 3207

5021 >> Mat([5021, 1]) >> 1
1363, 3658

5045 >> [5, 1; 1009, 1] >> 0
1478, 3567

5045 >> [5, 1; 1009, 1] >> 0
2487, 2558

5057 >> [13, 1; 389, 1] >> 0
1282, 3775

5057 >> [13, 1; 389, 1] >> 0
2449, 2608

5065 >> [5, 1; 1013, 1] >> 0
968, 4097

5065 >> [5, 1; 1013, 1] >> 0
1058, 4007

5069 >> [37, 1; 137, 1] >> 0
1881, 3188

5069 >> [37, 1; 137, 1] >> 0
1955, 3114

5077 >> Mat([5077, 1]) >> 1
858, 4219

5081 >> Mat([5081, 1]) >> 1
2412, 2669

5101 >> Mat([5101, 1]) >> 1
101, 5000

5105 >> [5, 1; 1021, 1] >> 0
647, 4458

5105 >> [5, 1; 1021, 1] >> 0
1668, 3437

5113 >> Mat([5113, 1]) >> 1
2025, 3088

5125 >> [5, 3; 41, 1] >> 0
1057, 4068

5125 >> [5, 3; 41, 1] >> 0
2182, 2943

5141 >> [53, 1; 97, 1] >> 0
507, 4634

5141 >> [53, 1; 97, 1] >> 0
560, 4581

5153 >> Mat([5153, 1]) >> 1
227, 4926

5161 >> [13, 1; 397, 1] >> 0
460, 4701

5161 >> [13, 1; 397, 1] >> 0
2319, 2842

5165 >> [5, 1; 1033, 1] >> 0
678, 4487

5165 >> [5, 1; 1033, 1] >> 0
1388, 3777

5185 >> [5, 1; 17, 1; 61, 1] >> 0
72, 5113

5185 >> [5, 1; 17, 1; 61, 1] >> 0
438, 4747

5185 >> [5, 1; 17, 1; 61, 1] >> 0
2002, 3183

5185 >> [5, 1; 17, 1; 61, 1] >> 0
2512, 2673

5189 >> Mat([5189, 1]) >> 1
2446, 2743

5197 >> Mat([5197, 1]) >> 1
1969, 3228

5209 >> Mat([5209, 1]) >> 1
2098, 3111

5213 >> [13, 1; 401, 1] >> 0
421, 4792

5213 >> [13, 1; 401, 1] >> 0
2426, 2787

5233 >> Mat([5233, 1]) >> 1
2253, 2980

5237 >> Mat([5237, 1]) >> 1
369, 4868

5245 >> [5, 1; 1049, 1] >> 0
623, 4622

5245 >> [5, 1; 1049, 1] >> 0
1672, 3573

5249 >> [29, 1; 181, 1] >> 0
162, 5087

5249 >> [29, 1; 181, 1] >> 0
1467, 3782

5261 >> Mat([5261, 1]) >> 1
827, 4434

5273 >> Mat([5273, 1]) >> 1
944, 4329

5281 >> Mat([5281, 1]) >> 1
1673, 3608

5297 >> Mat([5297, 1]) >> 1
2313, 2984

5305 >> [5, 1; 1061, 1] >> 0
103, 5202

5305 >> [5, 1; 1061, 1] >> 0
958, 4347

5309 >> Mat([5309, 1]) >> 1
1804, 3505

5317 >> [13, 1; 409, 1] >> 0
1084, 4233

5317 >> [13, 1; 409, 1] >> 0
1370, 3947

5321 >> [17, 1; 313, 1] >> 0
914, 4407

5321 >> [17, 1; 313, 1] >> 0
2529, 2792

5329 >> Mat([73, 2]) >> 0
776, 4553

5333 >> Mat([5333, 1]) >> 1
2630, 2703

5345 >> [5, 1; 1069, 1] >> 0
1318, 4027

5345 >> [5, 1; 1069, 1] >> 0
2387, 2958

5353 >> [53, 1; 101, 1] >> 0
394, 4959

5353 >> [53, 1; 101, 1] >> 0
818, 4535

5365 >> [5, 1; 29, 1; 37, 1] >> 0
302, 5063

5365 >> [5, 1; 29, 1; 37, 1] >> 0
882, 4483

5365 >> [5, 1; 29, 1; 37, 1] >> 0
2337, 3028

5365 >> [5, 1; 29, 1; 37, 1] >> 0
2448, 2917

5381 >> Mat([5381, 1]) >> 1
1739, 3642

5389 >> [17, 1; 317, 1] >> 0
837, 4552

5389 >> [17, 1; 317, 1] >> 0
2333, 3056

5393 >> Mat([5393, 1]) >> 1
665, 4728

5413 >> Mat([5413, 1]) >> 1
429, 4984

5417 >> Mat([5417, 1]) >> 1
368, 5049

5429 >> [61, 1; 89, 1] >> 0
233, 5196

5429 >> [61, 1; 89, 1] >> 0
1636, 3793

5437 >> Mat([5437, 1]) >> 1
630, 4807

5441 >> Mat([5441, 1]) >> 1
2452, 2989

5449 >> Mat([5449, 1]) >> 1
635, 4814

5465 >> [5, 1; 1093, 1] >> 0
563, 4902

5465 >> [5, 1; 1093, 1] >> 0
1623, 3842

5473 >> [13, 1; 421, 1] >> 0
450, 5023

5473 >> [13, 1; 421, 1] >> 0
1292, 4181

5477 >> Mat([5477, 1]) >> 1
74, 5403

5485 >> [5, 1; 1097, 1] >> 0
1438, 4047

5485 >> [5, 1; 1097, 1] >> 0
1853, 3632

5501 >> Mat([5501, 1]) >> 1
1115, 4386

5513 >> [37, 1; 149, 1] >> 0
105, 5408

5513 >> [37, 1; 149, 1] >> 0
1893, 3620

5521 >> Mat([5521, 1]) >> 1
765, 4756

5525 >> [5, 2; 13, 1; 17, 1] >> 0
268, 5257

5525 >> [5, 2; 13, 1; 17, 1] >> 0
1568, 3957

5525 >> [5, 2; 13, 1; 17, 1] >> 0
1968, 3557

5525 >> [5, 2; 13, 1; 17, 1] >> 0
2257, 3268

5545 >> [5, 1; 1109, 1] >> 0
1463, 4082

5545 >> [5, 1; 1109, 1] >> 0
2572, 2973

5557 >> Mat([5557, 1]) >> 1
2478, 3079

5569 >> Mat([5569, 1]) >> 1
973, 4596

5573 >> Mat([5573, 1]) >> 1
2017, 3556

5581 >> Mat([5581, 1]) >> 1
1437, 4144

5585 >> [5, 1; 1117, 1] >> 0
903, 4682

5585 >> [5, 1; 1117, 1] >> 0
2448, 3137

5597 >> [29, 1; 193, 1] >> 0
853, 4744

5597 >> [29, 1; 193, 1] >> 0
2042, 3555

5617 >> [41, 1; 137, 1] >> 0
237, 5380

5617 >> [41, 1; 137, 1] >> 0
2018, 3599

5629 >> [13, 1; 433, 1] >> 0
1045, 4584

5629 >> [13, 1; 433, 1] >> 0
2777, 2852

5641 >> Mat([5641, 1]) >> 1
1429, 4212

5645 >> [5, 1; 1129, 1] >> 0
168, 5477

5645 >> [5, 1; 1129, 1] >> 0
1297, 4348

5653 >> Mat([5653, 1]) >> 1
310, 5343

5657 >> Mat([5657, 1]) >> 1
1670, 3987

5669 >> Mat([5669, 1]) >> 1
1046, 4623

5689 >> Mat([5689, 1]) >> 1
2124, 3565

5693 >> Mat([5693, 1]) >> 1
1193, 4500

5701 >> Mat([5701, 1]) >> 1
385, 5316

5713 >> [29, 1; 197, 1] >> 0
1984, 3729

5713 >> [29, 1; 197, 1] >> 0
2772, 2941

5717 >> Mat([5717, 1]) >> 1
2416, 3301

5725 >> [5, 2; 229, 1] >> 0
107, 5618

5725 >> [5, 2; 229, 1] >> 0
2168, 3557

5729 >> [17, 1; 337, 1] >> 0
863, 4866

5729 >> [17, 1; 337, 1] >> 0
1874, 3855

5737 >> Mat([5737, 1]) >> 1
1126, 4611

5741 >> Mat([5741, 1]) >> 1
2378, 3363

5749 >> Mat([5749, 1]) >> 1
806, 4943

5765 >> [5, 1; 1153, 1] >> 0
1013, 4752

5765 >> [5, 1; 1153, 1] >> 0
1293, 4472

5777 >> [53, 1; 109, 1] >> 0
76, 5701

5777 >> [53, 1; 109, 1] >> 0
2256, 3521

5785 >> [5, 1; 13, 1; 89, 1] >> 0
1123, 4662

5785 >> [5, 1; 13, 1; 89, 1] >> 0
1568, 4217

5785 >> [5, 1; 13, 1; 89, 1] >> 0
1903, 3882

5785 >> [5, 1; 13, 1; 89, 1] >> 0
2348, 3437

5801 >> Mat([5801, 1]) >> 1
1145, 4656

5809 >> [37, 1; 157, 1] >> 0
1856, 3953

5809 >> [37, 1; 157, 1] >> 0
2226, 3583

5813 >> Mat([5813, 1]) >> 1
796, 5017

5821 >> Mat([5821, 1]) >> 1
1242, 4579

5825 >> [5, 2; 233, 1] >> 0
843, 4982

5825 >> [5, 2; 233, 1] >> 0
2707, 3118

5837 >> [13, 1; 449, 1] >> 0
382, 5455

5837 >> [13, 1; 449, 1] >> 0
2761, 3076

5849 >> Mat([5849, 1]) >> 1
2839, 3010

5857 >> Mat([5857, 1]) >> 1
1310, 4547

5861 >> Mat([5861, 1]) >> 1
754, 5107

5869 >> Mat([5869, 1]) >> 1
1042, 4827

5881 >> Mat([5881, 1]) >> 1
1098, 4783

5897 >> Mat([5897, 1]) >> 1
543, 5354

5905 >> [5, 1; 1181, 1] >> 0
243, 5662

5905 >> [5, 1; 1181, 1] >> 0
938, 4967

5917 >> [61, 1; 97, 1] >> 0
172, 5745

5917 >> [61, 1; 97, 1] >> 0
560, 5357

5933 >> [17, 1; 349, 1] >> 0
2656, 3277

5933 >> [17, 1; 349, 1] >> 0
2928, 3005

5941 >> [13, 1; 457, 1] >> 0
109, 5832

5941 >> [13, 1; 457, 1] >> 0
2176, 3765

5945 >> [5, 1; 29, 1; 41, 1] >> 0
278, 5667

5945 >> [5, 1; 29, 1; 41, 1] >> 0
1467, 4478

5945 >> [5, 1; 29, 1; 41, 1] >> 0
2018, 3927

5945 >> [5, 1; 29, 1; 41, 1] >> 0
2738, 3207

5953 >> Mat([5953, 1]) >> 1
2403, 3550

5965 >> [5, 1; 1193, 1] >> 0
1007, 4958

5965 >> [5, 1; 1193, 1] >> 0
2572, 3393

5981 >> Mat([5981, 1]) >> 1
1317, 4664

5989 >> [53, 1; 113, 1] >> 0
1454, 4535

5989 >> [53, 1; 113, 1] >> 0
1567, 4422

5993 >> [13, 1; 461, 1] >> 0
970, 5023

5993 >> [13, 1; 461, 1] >> 0
2257, 3736

6001 >> [17, 1; 353, 1] >> 0
395, 5606

6001 >> [17, 1; 353, 1] >> 0
1101, 4900

6005 >> [5, 1; 1201, 1] >> 0
1152, 4853

6005 >> [5, 1; 1201, 1] >> 0
2353, 3652

6025 >> [5, 2; 241, 1] >> 0
418, 5607

6025 >> [5, 2; 241, 1] >> 0
1382, 4643

6029 >> Mat([6029, 1]) >> 1
1801, 4228

6037 >> Mat([6037, 1]) >> 1
2652, 3385

6053 >> Mat([6053, 1]) >> 1
2832, 3221

6065 >> [5, 1; 1213, 1] >> 0
718, 5347

6065 >> [5, 1; 1213, 1] >> 0
1708, 4357

6073 >> Mat([6073, 1]) >> 1
2524, 3549

6085 >> [5, 1; 1217, 1] >> 0
78, 6007

6085 >> [5, 1; 1217, 1] >> 0
2512, 3573

6089 >> Mat([6089, 1]) >> 1
455, 5634

6101 >> Mat([6101, 1]) >> 1
247, 5854

6109 >> [41, 1; 149, 1] >> 0
1385, 4724

6109 >> [41, 1; 149, 1] >> 0
2428, 3681

6113 >> Mat([6113, 1]) >> 1
1089, 5024

6121 >> Mat([6121, 1]) >> 1
2583, 3538

6133 >> Mat([6133, 1]) >> 1
865, 5268

6145 >> [5, 1; 1229, 1] >> 0
597, 5548

6145 >> [5, 1; 1229, 1] >> 0
632, 5513

6161 >> [61, 1; 101, 1] >> 0
111, 6050

6161 >> [61, 1; 101, 1] >> 0
2939, 3222

6173 >> Mat([6173, 1]) >> 1
2447, 3726

6185 >> [5, 1; 1237, 1] >> 0
1783, 4402

6185 >> [5, 1; 1237, 1] >> 0
1928, 4257

6197 >> Mat([6197, 1]) >> 1
2007, 4190

6205 >> [5, 1; 17, 1; 73, 1] >> 0
557, 5648

6205 >> [5, 1; 17, 1; 73, 1] >> 0
922, 5283

6205 >> [5, 1; 17, 1; 73, 1] >> 0
1798, 4407

6205 >> [5, 1; 17, 1; 73, 1] >> 0
2163, 4042

6217 >> Mat([6217, 1]) >> 1
2372, 3845

6221 >> Mat([6221, 1]) >> 1
1121, 5100

6229 >> Mat([6229, 1]) >> 1
1451, 4778

6245 >> [5, 1; 1249, 1] >> 0
1913, 4332

6245 >> [5, 1; 1249, 1] >> 0
3083, 3162

6253 >> [13, 2; 37, 1] >> 0
746, 5507

6253 >> [13, 2; 37, 1] >> 0
2436, 3817

6257 >> Mat([6257, 1]) >> 1
1584, 4673

6269 >> Mat([6269, 1]) >> 1
1523, 4746

6277 >> Mat([6277, 1]) >> 1
1033, 5244

6301 >> Mat([6301, 1]) >> 1
2184, 4117

6305 >> [5, 1; 13, 1; 97, 1] >> 0
463, 5842

6305 >> [5, 1; 13, 1; 97, 1] >> 0
798, 5507

6305 >> [5, 1; 13, 1; 97, 1] >> 0
1477, 4828

6305 >> [5, 1; 13, 1; 97, 1] >> 0
2738, 3567

6317 >> Mat([6317, 1]) >> 1
1963, 4354

6329 >> Mat([6329, 1]) >> 1
2219, 4110

6337 >> Mat([6337, 1]) >> 1
178, 6159

6341 >> [17, 1; 373, 1] >> 0
642, 5699

6341 >> [17, 1; 373, 1] >> 0
2342, 3999

6353 >> Mat([6353, 1]) >> 1
1392, 4961

6361 >> Mat([6361, 1]) >> 1
1751, 4610

6373 >> Mat([6373, 1]) >> 1
1879, 4494

6385 >> [5, 1; 1277, 1] >> 0
113, 6272

6385 >> [5, 1; 1277, 1] >> 0
2667, 3718

6389 >> Mat([6389, 1]) >> 1
2092, 4297

6397 >> Mat([6397, 1]) >> 1
1302, 5095

6401 >> [37, 1; 173, 1] >> 0
80, 6321

6401 >> [37, 1; 173, 1] >> 0
253, 6148

6409 >> [13, 1; 17, 1; 29, 1] >> 0
684, 5725

6409 >> [13, 1; 17, 1; 29, 1] >> 0
1815, 4594

6409 >> [13, 1; 17, 1; 29, 1] >> 0
2163, 4246

6409 >> [13, 1; 17, 1; 29, 1] >> 0
3115, 3294

6421 >> Mat([6421, 1]) >> 1
825, 5596

6425 >> [5, 2; 257, 1] >> 0
2843, 3582

6425 >> [5, 2; 257, 1] >> 0
3068, 3357

6437 >> [41, 1; 157, 1] >> 0
1385, 5052

6437 >> [41, 1; 157, 1] >> 0
2697, 3740

6445 >> [5, 1; 1289, 1] >> 0
1768, 4677

6445 >> [5, 1; 1289, 1] >> 0
3057, 3388

6449 >> Mat([6449, 1]) >> 1
1854, 4595

6469 >> Mat([6469, 1]) >> 1
2977, 3492

6473 >> Mat([6473, 1]) >> 1
1808, 4665

6481 >> Mat([6481, 1]) >> 1
729, 5752

6485 >> [5, 1; 1297, 1] >> 0
1333, 5152

6485 >> [5, 1; 1297, 1] >> 0
2558, 3927

6497 >> [73, 1; 89, 1] >> 0
411, 6086

6497 >> [73, 1; 89, 1] >> 0
1725, 4772

6505 >> [5, 1; 1301, 1] >> 0
1352, 5153

6505 >> [5, 1; 1301, 1] >> 0
2653, 3852

6521 >> Mat([6521, 1]) >> 1
2364, 4157

6529 >> Mat([6529, 1]) >> 1
2311, 4218

6553 >> Mat([6553, 1]) >> 1
3186, 3367

6565 >> [5, 1; 13, 1; 101, 1] >> 0
697, 5868

6565 >> [5, 1; 13, 1; 101, 1] >> 0
798, 5767

6565 >> [5, 1; 13, 1; 101, 1] >> 0
1828, 4737

6565 >> [5, 1; 13, 1; 101, 1] >> 0
3242, 3323

6569 >> Mat([6569, 1]) >> 1
3038, 3531

6577 >> Mat([6577, 1]) >> 1
1624, 4953

6581 >> Mat([6581, 1]) >> 1
2727, 3854

6605 >> [5, 1; 1321, 1] >> 0
257, 6348

6605 >> [5, 1; 1321, 1] >> 0
1578, 5027

6613 >> [17, 1; 389, 1] >> 0
115, 6498

6613 >> [17, 1; 389, 1] >> 0
1441, 5172

6617 >> [13, 1; 509, 1] >> 0
1828, 4789

6617 >> [13, 1; 509, 1] >> 0
2244, 4373

6625 >> [5, 3; 53, 1] >> 0
182, 6443

6625 >> [5, 3; 53, 1] >> 0
818, 5807

6637 >> Mat([6637, 1]) >> 1
2828, 3809

6641 >> [29, 1; 229, 1] >> 0
336, 6305

6641 >> [29, 1; 229, 1] >> 0
1496, 5145

6649 >> [61, 1; 109, 1] >> 0
294, 6355

6649 >> [61, 1; 109, 1] >> 0
621, 6028

6653 >> Mat([6653, 1]) >> 1
752, 5901

6661 >> Mat([6661, 1]) >> 1
658, 6003

6673 >> Mat([6673, 1]) >> 1
2437, 4236

6689 >> Mat([6689, 1]) >> 1
2759, 3930

6697 >> [37, 1; 181, 1] >> 0
524, 6173

6697 >> [37, 1; 181, 1] >> 0
1067, 5630

6701 >> Mat([6701, 1]) >> 1
1721, 4980

6709 >> Mat([6709, 1]) >> 1
2150, 4559

6725 >> [5, 2; 269, 1] >> 0
82, 6643

6725 >> [5, 2; 269, 1] >> 0
1532, 5193

6733 >> Mat([6733, 1]) >> 1
2217, 4516

6737 >> Mat([6737, 1]) >> 1
2393, 4344

6749 >> [17, 1; 397, 1] >> 0
1254, 5495

6749 >> [17, 1; 397, 1] >> 0
2716, 4033

6757 >> [29, 1; 233, 1] >> 0
2419, 4338

6757 >> [29, 1; 233, 1] >> 0
3173, 3584

6761 >> Mat([6761, 1]) >> 1
1775, 4986

6773 >> [13, 1; 521, 1] >> 0
2319, 4454

6773 >> [13, 1; 521, 1] >> 0
2891, 3882

6781 >> Mat([6781, 1]) >> 1
995, 5786

6793 >> Mat([6793, 1]) >> 1
709, 6084

6805 >> [5, 1; 1361, 1] >> 0
747, 6058

6805 >> [5, 1; 1361, 1] >> 0
2108, 4697

6817 >> [17, 1; 401, 1] >> 0
421, 6396

6817 >> [17, 1; 401, 1] >> 0
1985, 4832

6829 >> Mat([6829, 1]) >> 1
1596, 5233

6833 >> Mat([6833, 1]) >> 1
1307, 5526

6841 >> Mat([6841, 1]) >> 1
1625, 5216

6845 >> [5, 1; 37, 2] >> 0
117, 6728

6845 >> [5, 1; 37, 2] >> 0
1252, 5593

6857 >> Mat([6857, 1]) >> 1

[/CODE]

Exercise for the readers from the previous centuries:
Come up with a routine to generate n's with complete set of known (a, b) pairs to determine if they have only-one or more prime factors.:smile:

 Nick 2021-01-12 09:32

Try Gaussian integers!

 a1call 2021-01-13 07:53

While the concept does not apply to Mersenne numbers since
valuation(Mn,2)==1

It does apply to Fermat numbers greater than F0.

The following code will find the 1st (a, b) pair for Fermat numbers greater than F0 virtually-instantly:

[CODE]

for(n=1,9,{
fermatNumber = 2^(2^n)+1;
print("\nfermatNumber = ",fermatNumber );
a = sqrtint(fermatNumber -1);print("a = ",a );
b=fermatNumber -a;;print("b = ",b );
m=lift(Mod(a*b,fermatNumber ););
print("F",n," >>-->> m = ",m);
})

[/CODE]

Output:

[CODE]
fermatNumber = 5
a = 2
b = 3
F1 >>-->> m = 1

fermatNumber = 17
a = 4
b = 13
F2 >>-->> m = 1

fermatNumber = 257
a = 16
b = 241
F3 >>-->> m = 1

fermatNumber = 65537
a = 256
b = 65281
F4 >>-->> m = 1

fermatNumber = 4294967297
a = 65536
b = 4294901761
F5 >>-->> m = 1

fermatNumber = 18446744073709551617
a = 4294967296
b = 18446744069414584321
F6 >>-->> m = 1

fermatNumber = 340282366920938463463374607431768211457
a = 18446744073709551616
b = 340282366920938463444927863358058659841
F7 >>-->> m = 1

fermatNumber = 115792089237316195423570985008687907853269984665640564039457584007913129639937
a = 340282366920938463463374607431768211456
b = 115792089237316195423570985008687907852929702298719625575994209400481361428481
F8 >>-->> m = 1

fermatNumber = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
a = 115792089237316195423570985008687907853269984665640564039457584007913129639936
b = 13407807929942597099574024998205846127479365820592393377723561443721764030073431184712636981971479856705023170278632780869088242247907112362425735876444161
F9 >>-->> m = 1
[/CODE]

For F1 to F4 there are no other positive integer pairs (a, b) other than listed.
For F5 however there is a 2nd pair:

[CODE]

\\DTC-120-A From Rashid Naimi - 1/13/2321 BC

F5 = 2^(2^5)+1
a = 46837383
b = F5-a
(a*b-1)/F5

[/CODE]

Output:

[CODE]

(02:48) gp > F5 = 2^(2^5)+1
%31 = 4294967297
(02:48) gp > a = 46837383
%32 = 46837383
(02:48) gp > b = F5-a
%33 = 4248129914
(02:48) gp > (a*b-1)/F5
%34 = 46326613

[/CODE]

Unfortunately I have no clue how to find the secondary (a, b) pairs without [STRIKE]Bruce-Lee [/STRIKE] Brute-Force.

Thank you for your time.
:smile:

 All times are UTC. The time now is 00:21.