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## Abstract

We can determinate if one number is prime with the modulo operation.
Primality Test Formula
5th December 2020
Pedro Hugo García Peláez
© Pedro Hugo García Peláez, 2020
This primality test it´s a consequence of the formula that finds prime numbers of the
form (p) mod (m) = (n)
First we need to know the quotient and remainder of our number when it´s divided by
any number, for example we are going to test 294662569 if we divided this number
by any number for example 3467876 we can deternine it´s remainder. In this case
294662569 mod 3467876 = 3360985
In this case (m)= 34687876 and (n)= 3360985
Now we use the formula (-2 + 3 x^(2/(m)))/(3 x^((n)/(m))) to find primes of the form
(p) mod (m) = (n)
Where (p) is a prime number.
And we put our data into the formula and we calculate the n(th) derivative.
d^90/dx^90((-2 + 3 x^(2/3467876))/(3 x^(3360985/3467876))) =
(51512098799692290321927489139786554202179216012422191774090244458134
14020550990632793926905635551446544064220332984326119946797210677486
3927364349365234375
(99395417620796243691243348705767899440899881826096321219651199266280
42736497564463817637726338748445850782337116094292835021771511060772
56875353956805965831503284102311895741206050127987009253755490644783
33490198622353187361896570826459860117531669959101997423651213088842
15528528916794116864863913775208439200732695430253481981217403258946
56944922361917669922388025085683452879904376467569986406600079230792
90806103592377736728567352044518791951752596227259887088866537255666
40946654190497161263724942445156490241029795302950331059215235458477
6621016655305474806766773179 x^(1/1733938) -
66263807950864644385798371977219380654360166647622133846018602129290
30249179446418772624689300185032242764155198792869411806289769137935
57875466633467191742022103799779850248456730863910032516864108760207
12743147728164811407050330192090388410708473781991772534403983001063
67497907486060587219067936170802400819462050289127608619428992411257
17944758046282491881994302293275834850977089564668582750909258744191
59167724272132652667606894631235120768072983967298347203853781562653
43273756260646298027298499535799391438480492262616266884801778869115
4791249421647995150797566242))/
(40338561855783284373959429860555039804007110724901891226753539432893
28889286631757823178423594994524019059751765076549601367278423040616
96552339314370038157068492654434004574746192903234824821447684459438
130135333357495545304323783018452711038264821994019954111445179735321
92017470793244625348120874818786664626932538905423842743558651417073
84032820332582604226795388732413397447319847974240487459538708480129
63580081487298659969924106740804646030923331661073855495059000315069
933712161020034150280322703879950752205524169267751156366111607966625
0409809797453976049917135686761657323749376 x^(315469825/3467876))
You need to increase the order of the derivative until the bound will be higher than
our number in this case the bound is 315469825 (the number in green) and our
formula find prime numbers of this form, strictly under this number (315469825)
If we divide the number in yellow by our number we have:
66263807950864644385798371977219380654360166647622133846018602129290
30249179446418772624689300185032242764155198792869411806289769137935
57875466633467191742022103799779850248456730863910032516864108760207
12743147728164811407050330192090388410708473781991772534403983001063
67497907486060587219067936170802400819462050289127608619428992411257
17944758046282491881994302293275834850977089564668582750909258744191
59167724272132652667606894631235120768072983967298347203853781562653
43273756260646298027298499535799391438480492262616266884801778869115
4791249421647995150797566242/ 294662569
224,880,303,513,761,343,694,109,895,502,945,203,244,868,765,966,749,356,0
94,898,507,890,529,870,768,195,414,015,162,023,830,050,332,325,812,450,07
7,389,804,790,977,923,285,842,863,047,147,147,703,537,191,295,587,361,759
,470,660,913,316,319,747,388,760,639,246,347,153,176,122,424,792,027,672,
885,072,315,819,631,947,851,546,453,033,815,746,622,824,034,304,136,755,4
05,058,166,246,442,629,062,530,713,590,649,541,220,216,887,782,839,109,75
5,708,143,956,933,309,964,472,033,939,228,295,502,403,386,646,252,641,980
,313,647,704,181,561,404,476,926,145,199,406,960,076,604,326,377,452,547,
205,345,135,211,198,023,927,997,464,292,051,165,322,707,607,163,387,158,8
90,456,103,351,756,204,617,848,169,655,461,609,271,614,060,314,159,396,30
6,795,321,695,560,018
As the resulting number is an integer we can determinate that 294662569 is prime.
For curiosity I will show you the factorization of the number in yellow of our
formula.
2 * 11 * 23 * 29 * 43 * 71 * 73 * 101 * 107 * 127 * 137 * 163 * 167 * 197 * 229 *
233 * 257 * 269 * 281 * 283 * 347 * 401 * 439 * 443 * 449 * 499 * 523 * 593 *
659 * 719 * 773 * 829 * 881 * 947 * 1051 * 1481 * 1747 * 1861 * 1979 * 2053 *
2297 * 2437 * 2477 * 2621 * 2699 * 2879 * 3257 * 3413 * 3677 * 4483 * 4657
* 4691 * 5119 * 7879 * 8431 * 9221 * 9721 * 9929 * 10993 * 12197 * 14153 *
15277 * 16087 * 17159 * 17257 * 17291 * 18461 * 21221 * 36013 * 39541 *
42557 * 52517 * 66103 * 81197 * 119489 * 149423 * 163109 * 276277 *
389713 * 393073 * 493931 * 712493 * 732967 * 741467 * 807011 * 885721 *
1 002853 * 1 275863 * 1 437797 * 2 017019 * 2 391737 * 2 535983 * 3
665581 * 6 003859 * 6 425071 * 9 150917 * 12 318349 * 12 939611 * 13
701277 * 17 864677 * 29 657729 * 31 103993 * 43 890799 * 56 158213 * 65
782753 * 75 101683 * 76 186381 * 114 333017 * 128 204521 * 145 543901 *
149 011777 * 169 819033 * 218 369297 * 221 837173 * 242 644429 * 253
048057 * 259 983809 * 270 387437 * 273 855313 * 284 258941 * 294
662569 * 301 598321
We can see that our number is the number in blue.
-Conclusions-
I did all the process in 30 seconds with a program that solve n(th) derivatives and a
big number calculator.
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