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## Abstract

We can find primes of the form prime(a)+prime(b)+1=prime(c) and prime(b)-prime(a)-1=prime (c) With the help of the Sophie Germain primes.
Primes of the form Prime(a)+Prime(b)+1=Prime(c)
27th December 2020
Pedro Hugo García Peláez
© Pedro Hugo García Peláez, 2020
The formula that we are going to use for find prime numbers of the form prime(a)
+prime(b)+1 is:
x^(4/(2*m)) – 3x^(2/m))
Is similar to the formula to find primes with a difference of (m) units. (prime(a)-
prime(b)=m
But now we have (2*m) in the first term of he formula instead of (m) and now (m)
have a important characteristic , (m) must be a Sophie Germain prime. We will see
after the relation with safe Sophie Germain´s primes.
Sophie Germain primes are primes were (2*Sophie Germain prime)+1 = other prime.
We use the nth derivative like always and we are going to use 284093 like Sophie
Germain prime.
60th derivative x^(4/(2*284093)) – 3x^(2/284093))
d^60/dx^60(x^(4/(2 284093)) - 3 x^(2/284093)) =
31494939259869510771685297980045311485493340486678040070695269330595
92571238654829191292663781350345114279098330559284243382285643200812
97913122605217724557413605189418850583902500241674970022614438221764
46309145019393918671553592468500982450266909983635527326500473872743
36363235794145738290910301086600624411475708364970107497270136107150
939074966622849661693504064970576970074608893952000000000000000/
(16129911147091475442878925225940861634220568424542112159328014929686
449131681969172169315111800850685318002816799294932351195235388202867
53675494001848332684311350229639331714802722143242662640219302166354
31261916943396823377651171457979287740260902317187100867008125266701
2798856934948096239623148179578361081676708809308558001
x^(17045578/284093))
I show only the last five primes of the factorization of the numerator (the number in
yellow).
3125021* 3503813* 9943253* 13352369* 14488741
We can find the relation
3503813 + 9943253 + 1 = 13447067
Where 13447067 is a prime number
The problem that don´t happen in my others formulas is that I didn´t find a form to
extract easily 3503813 and 9943253 because there isn´t and apparent relation
between this primes from our formula. And we need a machine to factorize the last 5
prime numbers of the factorization of our number.
With bigger derivatives we obtain bigger primes of the form prime(a)+prime(b)
+1=prime(c)
The success to find this relation between the last 5 numbers of the factorization is
about of 80% maybe a little bit more. In physics I consider a pattern any formula with
80% of success.
Now we are going with the safe Sophie Germain primes.
A safe Germain prime is of the form 2*(Sophie Germain prime)+1
I am going to test a safe sophie germain prime:
2*9520253+1= 19040507 where 9520253 is a Sophie Germain prime
We introduce this number into the formula
d^50/dx^50(x^(4/(2 19040507)) - 3 x^(2/19040507)) =
123143520261585739440704676027110455367111415568237890597731686381151
05639588012546449188564948630848512749493054523420346157674595786337
56295200921247320597336765729724444560591497693780637405348199878566
34590052752350110989520760790249888109433089529630965068683534564172
10956553748865746533754058629870387022595640906327914925655840207186
30181619188929161239709270292603533785569136427349326966020605214720
000000000000/
(96366351742149866028630509449149357300930890680478393443559694154693
88200891196418342645935817571507065239466830640727397252588911343022
28485932753919437678230770969720237608394051257361490885655847868713
02351646120605698536343402412480909770370000334695713593183627981245
21876257067065759055202324423830931915349063136616507734015261406037
417898411450756559676249 x^(952025348/19040507))
The last 5 primes of the factorization of the numerator are:
71401901* 88855699* 234832919* 418891153* 894903827
We can see the relation
418891153-234832919 -1=184058233
Where 184058233 is a prime number
Now we subtract two consecutive primes Prime(b)-Prime(a)-1
Where Prime(b)>Prime (a) and both primes are consecutive primes in the
factorization of the last 5 primes of our number in yellow.