Primality Certificate for (9464^2521-1)/9463 | ||
| Andy Steward | 10,020 digits | 21 February 2010 |
|---|---|---|
| Originally by A.A.D.Steward 2010 | ||
This certificate uses a theorem of Coppersmith and Howgrave-Graham to prove an integer N prime by making use of a partial prime factorization of N-1.
As N is a Generalized Repunit, we make use of the algebraic factorization of N-1 to arrive at the following 26.441069% factorization of N-1:
| From | Factorisation |
|---|---|
| 9464 | 2 · 2 · 2 · 7 · 13 · 13 |
| Φ2 | 3 · 5 · 631 |
| Φ3 | 3373 · 26557 |
| Φ4 | 53 · 233 · 7253 |
| Φ5 | 11 · 61 · 3422191 · 3493961 |
| Φ6 | 3 · 457 · 65323 |
| Φ7 | 43 · 16711899900041318913067 |
| Φ8 | 11446289 · 700864753 |
| Φ9 | 2791 · 756289 · 340408792829719 |
| Φ10 | 5 · 281 · 5709219172541 |
| Φ12 | 37 · 673 · 18061 · 17837761 |
| Φ14 | 113 · 617 · 10304784062133016177 |
| Φ15 | 31 · p31 |
| Φ18 | 3 · 19 · 307 · 3547 · 4933 · 2346728537557 |
| Φ20 | 78810001 · 816613424460673263835441 |
| Φ21 | 211 · 88411 · 948669793 · p32 |
| Φ24 | 193 · 297097 · 42655177 · 26313009897239473 |
| Φ28 | 29 · 421 · 6469 · 3685616916639815501 · 1773655774169309609041 |
| Φ30 | 347440547022601 · 185252142534499441 |
| Φ35 | 78121 · 584081 · 76388142203055205426665198571 · p56 |
| Φ36 | 109 · 2033497 · 292077613 · 2803490461 · 25561628173 · 111286088713 |
| Φ40 | 41 · 843875708521 · 102354585320865895201 · p31 |
| Φ42 | 127 · 415087 · 327001348944268153 · 29953685729424294387313 |
| Φ45 | 113761 · 3134881 · 3368791 · 2026974421 · 6572276520752257119271 · p47 |
| Φ56 | 504320170708052062448471442227761 · p63 |
| Φ60 | 150301 · 584319204421 · p47 |
| Φ63 | 1009 · 700561 · 67849808177971 · 30017622987197074657 · 677502141136520743499504571433 · p72 |
| Φ70 | 71 · 12384712271 · 15282214963499407351 · p65 |
| Φ72 | 73 · 1873 · 25057 · 1193473 · 813268665766529281 · 76120850244633143113 · p43 |
| Φ84 | 6217 · 38557 · 598333 · 17541487396273 · 90806289097758613 · 213259367914614667381 · p31 |
| Φ90 | 181 · 110161 · 217081 · 177270211 · p75 |
| Φ105 | 50746291 · 6180893198153479267088063641 · p156 |
| Φ120 | 13668481 · 39277681 · 88413896192274848410321 · 97722741730230091021681 · 146592839362552894226749017361 · p38 |
| Φ126 | 96769 · 149852527273 · 1417063822386093562585788740809 · 102072204632440555649945238010897 · p65 |
| Φ140 | 2801 · 1064281 · 3581889004471541 · 13515568007902849213361 · 846956999357977583853894761 · c117 |
| Φ168 | 337 · 804048673 · 16427405233 · c170 |
| Φ180 | 893341 · 1394954101 · p176 |
| Φ210 | 3397591 · 2746563961 · 226922028369502754521591 · 15155647953634655597007412994761 · c121 |
| Φ252 | 15877 · 2626063021 · 35799629293 · 871784684317 · 835118198047705069 · c233 |
| Φ280 | 39761 · 81813761 · 1649417281 · c360 |
| Φ315 | 34651 · 58858600209691470862980601 · c543 |
| Φ360 | 184321 · 189402121 · 516665149921 · 8584371401495761 · 30270186581402367914791123029601 · c310 |
| Φ420 | 232864488361 · 144067073395706748725761 · c348 |
| Φ504 | 3049201 · 10839977569 · c557 |
| Φ630 | 158761 · 234361 · 2738378161 · 528867912137671 · 640057097255311 · 585128955100693982659715161 · c497 |
| Φ840 | 15121 · 86464561 · 88966575601 · c741 |
| Φ1260 | 2521 · 6301 · 7561 · 45361 · 9579190321 · 32596115047631101 · c1103 |
| Φ2520 | 2070515492641 · c2278 |
We need the product F of all the prime factors from this partial factorization:
| 570177 3133936450 5354317818 2601426314 2412302506 3178801839 3406240476 8303270783 1301737271 4887438715 3326413095 5011213101 0457139051 3295422071 1441581504 9102624085 5929735238 8224270401 |
| 226556 9891206892 7292497292 9683156613 0124968910 3049243697 7374652573 3715221915 6107041155 9621358267 9045406756 0198525136 4996754953 7596579114 4071788473 0689534811 |
| 34739 9780116906 9475983275 3076760909 9726318327 5020102644 4011474516 1407585191 |
| 14 1096948221 2141279767 3312156248 9892318139 5663818755 5410021463 7032171619 |
| 65613 4853297698 3744706333 5724432838 5639952358 6170856649 3072124481 |
| 19838 5018037881 3831323697 9742155962 7676313501 1381243351 2045615031 |
| 528 5513913555 9421412405 2929398816 2334388734 1961615138 3760471761 |
| 764681 9633580647 5598037479 6779852519 7264400212 6526704051 |
| 4716107 3319625439 1221096325 1025464784 6035163761 |
| 1665485 7999910742 6668890721 1701652481 4830493501 |
| 105 3061122764 8236904975 8079592220 8488081153 |
| 25228788 1857072777 8469420318 5291183041 |
| 504 3201707080 5206244847 1442227761 |
| 102 0722046324 4055564994 5238010897 |
| 30 2701865814 0236791479 1123029601 |
| 29 1707138819 5199359962 0314439337 |
| 15 1556479536 3465559700 7412994761 |
| 5 4711175150 4758543197 4700943817 |
| 2 0758227514 3530097114 1353124311 |
| 1 4170638223 8609356258 5788740809 |
| 1 1695695705 4342750094 2055503121 |
| 6775021411 3652074349 9504571433 |
| 1465928393 6255289422 6749017361 |
| 763881422 0305520542 6665198571 |
| 61808931 9815347926 7088063641 |
| 8469569 9935797758 3853894761 |
| 5851289 5510069398 2659715161 |
| 588586 0020969147 0862980601 |
| 8166 1342446067 3263835441 |
| 2269 2202836950 2754521591 |
| 1440 6707339570 6748725761 |
| 977 2274173023 0091021681 |
| 884 1389619227 4848410321 |
| 299 5368572942 4294387313 |
| 167 1189990004 1318913067 |
| 135 1556800790 2849213361 |
| 65 7227652075 2257119271 |
| 17 7365577416 9309609041 |
| 2 1325936791 4614667381 |
| 1 0235458532 0865895201 |
| 7612085024 4633143113 |
| 3001762298 7197074657 |
| 1528221496 3499407351 |
| 1030478406 2133016177 |
| 368561691 6639815501 |
| 83511819 8047705069 |
| 81326866 5766529281 |
| 32700134 8944268153 |
| 18525214 2534499441 |
| 9080628 9097758613 |
| 3259611 5047631101 |
| 2631300 9897239473 |
| 858437 1401495761 |
| 358188 9004471541 |
| 64005 7097255311 |
| 52886 7912137671 |
| 34744 0547022601 |
| 34040 8792829719 |
| 6784 9808177971 |
| 1754 1487396273 |
| 570 9219172541 |
| 234 6728537557 |
| 207 0515492641 |
| 87 1784684317 |
| 84 3875708521 |
| 58 4319204421 |
| 51 6665149921 |
| 23 2864488361 |
| 14 9852527273 |
| 11 1286088713 |
| 8 8966575601 |
| 3 5799629293 |
| 2 5561628173 |
| 1 6427405233 |
| 1 2384712271 |
| 1 0839977569 |
| 9579190321 |
| 2803490461 |
| 2746563961 |
| 2738378161 |
| 2626063021 |
| 2026974421 |
| 1649417281 |
| 1394954101 |
| 948669793 |
| 804048673 |
| 700864753 |
| 292077613 |
| 189402121 |
| 177270211 |
| 86464561 |
| 81813761 |
| 78810001 |
| 50746291 |
| 42655177 |
| 39277681 |
| 17837761 |
| 13668481 |
| 11446289 |
| 3493961 |
| 3422191 |
| 3397591 |
| 3368791 |
| 3134881 |
| 3049201 |
| 2033497 |
| 1193473 |
| 1064281 |
| 893341 |
| 756289 |
| 700561 |
| 598333 |
| 584081 |
| 415087 |
| 297097 |
| 234361 |
| 217081 |
| 184321 |
| 158761 |
| 150301 |
| 113761 |
| 110161 |
| 96769 |
| 88411 |
| 78121 |
| 65323 |
| 45361 |
| 39761 |
| 38557 |
| 34651 |
| 26557 |
| 25057 |
| 18061 |
| 15877 |
| 15121 |
| 7561 |
| 7253 |
| 6469 |
| 6301 |
| 6217 |
| 4933 |
| 3547 |
| 3373 |
| 2801 |
| 2791 |
| 2521 |
| 1873 |
| 1009 |
| 673 |
| 631 |
| 617 |
| 457 |
| 421 |
| 337 |
| 307 |
| 281 |
| 233 |
| 211 |
| 193 |
| 181 |
| 127 |
| 113 |
| 109 |
| 73 |
| 71 |
| 61 |
| 53 |
| 43 |
| 41 |
| 37 |
| 31 |
| 29 |
| 19 |
| 132 |
| 11 |
| 7 |
| 52 |
| 33 |
| 23 |
Note that all prime factors listed above have been proven. As primes of under 250 decimal digits can be verified in a few seconds, proof of their primality is not included here, in order to save space.
We set R = (N-1)/F. Note that GCD(F,R)=1 and Log(F)/Log(N) = 26.441069%
Next, we find an integer witness w such that for each prime factor p of N-1, w(N-1) ≡ 1 mod N and GCD(w(N-1)/p-1,N) = 1. In this case, w = 391 suffices.
Given such a witness, Pocklington's Theorem shows that every prime factor of N ≡ 1 (mod F). As F4>N, N can have no more than three prime factors.
As F2 < N < F3 and N ≡ 1 (mod F), we can let N = c2·F2 + c1·F + 1.
Brillhart, Lehmer and Selfridge's Theorem shows that N has exactly two prime factors if and only if c12-4·c2 is a perfect square.
Here, c12-4·c2 is ≡ 61 (mod 64) and therefore cannot be a square and this stage of the proof is passed.
We are left with two possibilities for N: either it has exactly three prime factors or it is prime. The non-existence of exactly three factors is demonstrated by the Theorem of Coppersmith and Howgrave-Graham, here performed by a Pari/GP script written by John Renze and David Broadhurst. Here is the stdout:
Welcome to the CHG primality prover!
------------------------------------
realprecision = 4257 significant digits (4250 digits displayed)
Input file is: IO\24F809D9.cin
Certificate file is: IO\24F809D9.chg
Found values of n, F and G.
Number to be tested has 10020 digits.
Modulus has 2650 digits.
Modulus is 26.44106944% of n.
NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!
Square test passed for F >> G. Using modified right endpoint.
Search for factors congruent to 1.
Running CHG with h = 18, u = 8. Right endpoint has 2073 digits.
Done! Time elapsed: 15793750ms.
Running CHG with h = 18, u = 8. Right endpoint has 2057 digits.
Done! Time elapsed: 25371734ms.
Running CHG with h = 18, u = 8. Right endpoint has 2040 digits.
Done! Time elapsed: 22037719ms.
Running CHG with h = 18, u = 8. Right endpoint has 2024 digits.
Done! Time elapsed: 15158469ms.
Running CHG with h = 18, u = 8. Right endpoint has 2006 digits.
Done! Time elapsed: 22335453ms.
Running CHG with h = 18, u = 8. Right endpoint has 1987 digits.
Done! Time elapsed: 24036562ms.
Running CHG with h = 17, u = 7. Right endpoint has 1967 digits.
Done! Time elapsed: 44411625ms.
Running CHG with h = 17, u = 7. Right endpoint has 1956 digits.
Done! Time elapsed: 46352641ms.
Running CHG with h = 17, u = 7. Right endpoint has 1944 digits.
Done! Time elapsed: 50156328ms.
Running CHG with h = 17, u = 7. Right endpoint has 1930 digits.
Done! Time elapsed: 53004625ms.
Running CHG with h = 17, u = 7. Right endpoint has 1914 digits.
Done! Time elapsed: 56149766ms.
Running CHG with h = 17, u = 7. Right endpoint has 1895 digits.
Done! Time elapsed: 14583640ms.
Running CHG with h = 17, u = 7. Right endpoint has 1874 digits.
Done! Time elapsed: 14091047ms.
Running CHG with h = 17, u = 7. Right endpoint has 1850 digits.
Done! Time elapsed: 15002281ms.
Running CHG with h = 15, u = 6. Right endpoint has 1823 digits.
Done! Time elapsed: 7333829ms.
Running CHG with h = 15, u = 6. Right endpoint has 1815 digits.
Done! Time elapsed: 7263328ms.
Running CHG with h = 15, u = 6. Right endpoint has 1806 digits.
Done! Time elapsed: 7132125ms.
Running CHG with h = 15, u = 6. Right endpoint has 1796 digits.
Done! Time elapsed: 5340765ms.
Running CHG with h = 15, u = 6. Right endpoint has 1784 digits.
Done! Time elapsed: 5544500ms.
Running CHG with h = 15, u = 6. Right endpoint has 1770 digits.
Done! Time elapsed: 5488657ms.
Running CHG with h = 15, u = 6. Right endpoint has 1754 digits.
Done! Time elapsed: 6938453ms.
Running CHG with h = 15, u = 6. Right endpoint has 1735 digits.
Done! Time elapsed: 7584156ms.
Running CHG with h = 15, u = 6. Right endpoint has 1712 digits.
Done! Time elapsed: 6738859ms.
Running CHG with h = 15, u = 6. Right endpoint has 1687 digits.
Done! Time elapsed: 7744219ms.
Running CHG with h = 15, u = 6. Right endpoint has 1656 digits.
Done! Time elapsed: 6178438ms.
Running CHG with h = 13, u = 5. Right endpoint has 1620 digits.
Done! Time elapsed: 1934718ms.
Running CHG with h = 13, u = 5. Right endpoint has 1607 digits.
Done! Time elapsed: 1970735ms.
Running CHG with h = 13, u = 5. Right endpoint has 1592 digits.
Done! Time elapsed: 2478750ms.
Running CHG with h = 13, u = 5. Right endpoint has 1573 digits.
Done! Time elapsed: 3030140ms.
Running CHG with h = 13, u = 5. Right endpoint has 1550 digits.
Done! Time elapsed: 4190625ms.
Running CHG with h = 13, u = 5. Right endpoint has 1522 digits.
Done! Time elapsed: 3489000ms.
Running CHG with h = 13, u = 5. Right endpoint has 1485 digits.
Done! Time elapsed: 2047047ms.
Running CHG with h = 11, u = 4. Right endpoint has 1445 digits.
Done! Time elapsed: 956625ms.
Running CHG with h = 11, u = 4. Right endpoint has 1435 digits.
Done! Time elapsed: 695656ms.
Running CHG with h = 12, u = 4. Right endpoint has 1421 digits.
Done! Time elapsed: 631422ms.
Running CHG with h = 11, u = 4. Right endpoint has 1392 digits.
Done! Time elapsed: 593891ms.
Running CHG with h = 11, u = 4. Right endpoint has 1364 digits.
Done! Time elapsed: 1381141ms.
Running CHG with h = 11, u = 4. Right endpoint has 1325 digits.
Done! Time elapsed: 1539140ms.
Running CHG with h = 11, u = 4. Right endpoint has 1280 digits.
Done! Time elapsed: 1748125ms.
Running CHG with h = 9, u = 3. Right endpoint has 1228 digits.
Done! Time elapsed: 260110ms.
Running CHG with h = 10, u = 3. Right endpoint has 1214 digits.
Done! Time elapsed: 288031ms.
Running CHG with h = 9, u = 3. Right endpoint has 1176 digits.
Done! Time elapsed: 182765ms.
Running CHG with h = 9, u = 3. Right endpoint has 1139 digits.
Done! Time elapsed: 318110ms.
Running CHG with h = 9, u = 3. Right endpoint has 1085 digits.
Done! Time elapsed: 321984ms.
Running CHG with h = 8, u = 2. Right endpoint has 1011 digits.
Done! Time elapsed: 105078ms.
Running CHG with h = 8, u = 2. Right endpoint has 998 digits.
Done! Time elapsed: 89453ms.
Running CHG with h = 7, u = 2. Right endpoint has 977 digits.
Done! Time elapsed: 78719ms.
Running CHG with h = 7, u = 2. Right endpoint has 962 digits.
Done! Time elapsed: 79297ms.
Running CHG with h = 7, u = 2. Right endpoint has 941 digits.
Done! Time elapsed: 79859ms.
Running CHG with h = 7, u = 2. Right endpoint has 910 digits.
Done! Time elapsed: 82688ms.
Running CHG with h = 7, u = 2. Right endpoint has 862 digits.
Done! Time elapsed: 85969ms.
Running CHG with h = 7, u = 2. Right endpoint has 791 digits.
Done! Time elapsed: 80797ms.
Running CHG with h = 7, u = 2. Right endpoint has 674 digits.
Done! Time elapsed: 82921ms.
Running CHG with h = 5, u = 1. Right endpoint has 477 digits.
Done! Time elapsed: 8469ms.
Running CHG with h = 5, u = 1. Right endpoint has 285 digits.
Done! Time elapsed: 11422ms.
A certificate has been saved to the file: IO\24F809D9.chg
Running David Broadhurst's verifier on the saved certificate...
Testing a PRP called "IO\24F809D9.cin".
Pol[1, 1] with [h, u]=[5, 1] has ratio=1.023077797 E-77 at X, ratio=1.269827630 E-361 at Y, witness=2.
Pol[2, 1] with [h, u]=[4, 1] has ratio=0.1017310548 at X, ratio=5.50507266 E-193 at Y, witness=3.
Pol[3, 1] with [h, u]=[7, 2] has ratio=0.591543987 at X, ratio=5.781954072 E-394 at Y, witness=3.
Pol[4, 1] with [h, u]=[7, 2] has ratio=3.046322358 E-118 at X, ratio=2.886322310 E-235 at Y, witness=3.
Pol[5, 1] with [h, u]=[7, 2] has ratio=0.0940057270 at X, ratio=5.350470802 E-143 at Y, witness=5.
Pol[6, 1] with [h, u]=[7, 2] has ratio=0.01357849329 at X, ratio=1.304267055 E-95 at Y, witness=2.
Pol[7, 1] with [h, u]=[7, 2] has ratio=0.1744338264 at X, ratio=5.181699930 E-64 at Y, witness=2.
Pol[8, 1] with [h, u]=[7, 2] has ratio=0.523706184 at X, ratio=5.584383566 E-43 at Y, witness=5.
Pol[9, 1] with [h, u]=[7, 2] has ratio=0.659452459 at X, ratio=8.01513700 E-29 at Y, witness=17.
Pol[10, 1] with [h, u]=[8, 2] has ratio=0.4511833422 at X, ratio=1.388997852 E-44 at Y, witness=2.
Pol[11, 1] with [h, u]=[8, 2] has ratio=0.4492020946 at X, ratio=7.42607992 E-26 at Y, witness=2.
Pol[12, 1] with [h, u]=[9, 3] has ratio=0.3339589668 at X, ratio=1.998562873 E-222 at Y, witness=5.
Pol[13, 1] with [h, u]=[9, 3] has ratio=9.37191336 E-56 at X, ratio=2.529642943 E-164 at Y, witness=2.
Pol[14, 1] with [h, u]=[9, 3] has ratio=1.890656719 E-37 at X, ratio=9.62848105 E-110 at Y, witness=2.
Pol[15, 1] with [h, u]=[10, 3] has ratio=0.4312208502 at X, ratio=6.65721209 E-116 at Y, witness=3.
Pol[16, 1] with [h, u]=[9, 3] has ratio=0.3701999379 at X, ratio=2.929170302 E-41 at Y, witness=7.
Pol[17, 1] with [h, u]=[11, 4] has ratio=0.1329637178 at X, ratio=2.861425048 E-209 at Y, witness=5.
Pol[18, 1] with [h, u]=[11, 4] has ratio=0.02652854121 at X, ratio=1.346012554 E-180 at Y, witness=5.
Pol[19, 1] with [h, u]=[11, 4] has ratio=3.462160685 E-40 at X, ratio=2.164660004 E-156 at Y, witness=11.
Pol[20, 1] with [h, u]=[11, 4] has ratio=1.258857243 E-29 at X, ratio=5.760599190 E-114 at Y, witness=7.
Pol[21, 1] with [h, u]=[12, 4] has ratio=0.01995287394 at X, ratio=6.90374695 E-117 at Y, witness=5.
Pol[22, 1] with [h, u]=[11, 4] has ratio=0.1389703497 at X, ratio=1.340782848 E-54 at Y, witness=11.
Pol[23, 1] with [h, u]=[11, 4] has ratio=0.4691259622 at X, ratio=7.16034351 E-44 at Y, witness=3.
Pol[24, 1] with [h, u]=[13, 5] has ratio=0.3193168381 at X, ratio=9.43372091 E-201 at Y, witness=3.
Pol[25, 1] with [h, u]=[13, 5] has ratio=0.3911194749 at X, ratio=2.611167972 E-182 at Y, witness=3.
Pol[26, 1] with [h, u]=[13, 5] has ratio=2.922277029 E-31 at X, ratio=8.06669018 E-144 at Y, witness=7.
Pol[27, 1] with [h, u]=[13, 5] has ratio=0.3533208358 at X, ratio=2.859806535 E-113 at Y, witness=2.
Pol[28, 1] with [h, u]=[13, 5] has ratio=0.05659375834 at X, ratio=1.413074745 E-94 at Y, witness=17.
Pol[29, 1] with [h, u]=[13, 5] has ratio=0.3319801767 at X, ratio=6.80972915 E-79 at Y, witness=5.
Pol[30, 1] with [h, u]=[13, 5] has ratio=0.003951810439 at X, ratio=7.83819582 E-66 at Y, witness=3.
Pol[31, 1] with [h, u]=[15, 6] has ratio=2.551469969 E-37 at X, ratio=6.208294720 E-217 at Y, witness=2.
Pol[32, 1] with [h, u]=[15, 6] has ratio=0.01341236499 at X, ratio=2.056406614 E-182 at Y, witness=13.
Pol[33, 1] with [h, u]=[15, 6] has ratio=0.3995895740 at X, ratio=1.902552083 E-156 at Y, witness=29.
Pol[34, 1] with [h, u]=[15, 6] has ratio=0.01322251299 at X, ratio=3.710567807 E-134 at Y, witness=5.
Pol[35, 1] with [h, u]=[15, 6] has ratio=0.02846552679 at X, ratio=3.894818802 E-115 at Y, witness=3.
Pol[36, 1] with [h, u]=[15, 6] has ratio=0.0982482712 at X, ratio=9.13538561 E-99 at Y, witness=5.
Pol[37, 1] with [h, u]=[15, 6] has ratio=0.3287412862 at X, ratio=8.92127513 E-85 at Y, witness=17.
Pol[38, 1] with [h, u]=[15, 6] has ratio=0.2318373530 at X, ratio=8.41445331 E-73 at Y, witness=3.
Pol[39, 1] with [h, u]=[15, 6] has ratio=0.1458260553 at X, ratio=1.565757320 E-62 at Y, witness=11.
Pol[40, 1] with [h, u]=[15, 6] has ratio=0.0735967851 at X, ratio=1.156359747 E-53 at Y, witness=2.
Pol[41, 1] with [h, u]=[15, 6] has ratio=0.0690534130 at X, ratio=3.727562612 E-46 at Y, witness=2.
Pol[42, 1] with [h, u]=[17, 7] has ratio=0.02099277131 at X, ratio=1.982515385 E-192 at Y, witness=5.
Pol[43, 1] with [h, u]=[17, 7] has ratio=0.0838866206 at X, ratio=1.517858886 E-168 at Y, witness=13.
Pol[44, 1] with [h, u]=[17, 7] has ratio=0.004670709894 at X, ratio=1.438310348 E-147 at Y, witness=3.
Pol[45, 1] with [h, u]=[17, 7] has ratio=0.644120718 at X, ratio=3.160720493 E-129 at Y, witness=5.
Pol[46, 1] with [h, u]=[17, 7] has ratio=0.04908918008 at X, ratio=4.023972363 E-113 at Y, witness=19.
Pol[47, 1] with [h, u]=[17, 7] has ratio=0.1402511560 at X, ratio=4.829129714 E-99 at Y, witness=3.
Pol[48, 1] with [h, u]=[17, 7] has ratio=0.1168952096 at X, ratio=7.858530382 E-87 at Y, witness=3.
Pol[49, 1] with [h, u]=[17, 7] has ratio=0.501113220 at X, ratio=5.141189938 E-76 at Y, witness=2.
Pol[50, 1] with [h, u]=[18, 8] has ratio=0.04889076156 at X, ratio=5.63248878 E-162 at Y, witness=3.
Pol[51, 1] with [h, u]=[18, 8] has ratio=0.4873766570 at X, ratio=1.926633049 E-152 at Y, witness=13.
Pol[52, 1] with [h, u]=[18, 8] has ratio=0.002104172452 at X, ratio=1.262888396 E-143 at Y, witness=3.
Pol[53, 1] with [h, u]=[18, 8] has ratio=0.2387755742 at X, ratio=3.612929760 E-135 at Y, witness=2.
Pol[54, 1] with [h, u]=[18, 8] has ratio=0.02058455563 at X, ratio=3.347834524 E-132 at Y, witness=3.
Pol[55, 1] with [h, u]=[18, 8] has ratio=3.698651356 E-18 at X, ratio=5.620896052 E-126 at Y, witness=3.
Validated in 35 sec.
Congratulations! n is prime!
The actual input file containing N and F and the output certificate are included in this file.