Content uploaded by Pedro Hugo García Peláez

Author content

All content in this area was uploaded by Pedro Hugo García Peláez on Sep 25, 2020

Content may be subject to copyright.

Content uploaded by Pedro Hugo García Peláez

Author content

All content in this area was uploaded by Pedro Hugo García Peláez on Sep 20, 2020

Content may be subject to copyright.

Content uploaded by Pedro Hugo García Peláez

Author content

All content in this area was uploaded by Pedro Hugo García Peláez on Aug 22, 2020

Content may be subject to copyright.

Conclusions

25th of September of 2020

We can find any prime of the form ((m)*(n))+1 just under of a number given.

Because that number given depends of the order of the derivative * The

coefficient (m)+1

Better with an example

If we want to search the prime of the form (199*n)+1 just under than 19,900

we use the derivative 100 in our formula.

100th derivative (-2/x^(1/199) + (4 x^(1/199))/1

d^100/dx^100(-2/x^(1/199) + 4 x^(1/199)) = -

(99364166484840112256708922645041637995448020289168347414383980959309

77604225271478451567997266651943887682613902625240630756400260258807

4672130259396141609459384320000000000000000000000000

(14124738050219081504823475627787268312082340753306253593396010464249

47134801096089880492729526564485794206039961909522697735028587280446

3187506556774268539415656610053022758274688636348233364040819

x^(2/199) +

74395815037759038466048139180476073256924007194049059793540321633040

15375586945896427977202396635263570951463768739415871568529686983470

388556886477432807087259005918095557574075248654438621000088))/

(76790525741798814147739722024656541832379779544942834289320631222830

12863136141226080950493607708198950026290145142696888920938866374089

05005446892190734624961059132500532096789390465051335714291661355241

05744097027866266597980001 x^(19901/199))

The exponent in blue is the number given.

We choose the number in yellow

The factorization is

2^3×3×19^2×23×41×59×61×73×83×151×181×193×211×227×229×233×239×263×271

×281×293×311×347×359×389×419×421×439×457×607×617×643×647×661×709×743

×787×797×821×853×929×977×1123×1277×1327×1433×1439×1493×1559×1811×183

1×2239×2389×2521×2687×2753×3433×3583×3821×3881×4229×4909×5573×6269×9

851×11941×13931×16319×17911

We can see that the bigger of it´s primes is 17911 and 17910/199=90 so (n)=90

and (199*90)+1=17911 and this is the biggest prime number of the form

199*(n)+1 just under 19901

To obtain this prime we have to divide the number in yellow from (i=19,900 to

i=1) The first integer is the prime we are looking for. In our case that integer

will be when we divide the number in yellow by 17910