^{A study of Integers Using Software Tools – I}

Cătălin Angelo Ioan¹, Alin Cristian Ioan²

Abstract. The paper reviews the Sophie German primes and then it introduces two new types related to them. Using software there will be determined the first sequences of such numbers.

Keywords: prime; Sophie Germain

1 Introduction

The prime number theory dates back to ancient times (see the Rhind papyrus or Euclid's Elements).

A number p￿N, p2 is called prime if its only positive divisors are 1 and p. The remarkable property of primes is that any nonzero natural number other than 1 can be written as a unique product (up to a permutation of factors) of prime numbers to various powers.

All over in this paper, the software presented was written in Wolfram Mathematica 9.0.

2 Sophie Germain Primes

A prime number p is called a Sophie Germain prime if 2p+1 is also prime. The number 2p+1 is called safe prime.

A prime number p5 is of the form 6k-1 or 6k+1, k1. If p=6k-1 then 2p+1=12k-1 and if p=6k+1 then 2p+1=12k+3=3(4k+1) which is not prime. Therefore, Sophie Germain primes numbers p5 are necessarily of the form p=6k-1.

Let note that p=2 and p=3 are Sophie Germain primes.

The software for determining the Sophie Germain primes and the related safe primes is:

Clear[“Global`*”];

(*Sophie Germain primes*)

limit=100;

For[k=1,k<limit,k++,If[PrimeQ[6*k-1]&& PrimeQ[12*k-1],Print[6*k-1,” “,12*k-1]]]

and we find (with the limit 599 for Sophie Germain primes ):

3 A New Type of Numbers: ,-primes

Based on the idea of Sophie Germain prime numbers we define a number p to be ,-prime if:
p, p+1, ²p+1, ..., ^p+1 are all primes.

Obvious if p2,  must be an even number, otherwise ^sp+1=even s= .

If p5, p=6k-1 that is p5 (mod 6) then p+15+1 (mod 6), ²p+15²+1 (mod 6),..., ^p+15^+1 (mod 6).

If =2 we have therefore: p+14+1 (mod 6), ²p+12²+1 (mod 6),..., ^p+152^^+1 (mod 6).

• If 0 (mod 6) we have: p+11 (mod 6), ²p+11 (mod 6), ^p+11 (mod 6).

• If 2 (mod 6) we have: p+15 (mod 6), ²p+13 (mod 6) therefore we cannot have 12+2,-primes if 2 and p=6k-1, k1.

• If 4 (mod 6) we have: p+13 (mod 6) ) therefore we cannot have 12+4,-primes if 1 and p=6k-1, k1.

If now p7, p=6k+1 that is p1 (mod 6) then p+1+1 (mod 6), ²p+1²+1 (mod 6),..., ^p+1^+1 (mod 6).

If =2 we have therefore: p+12+1 (mod 6), ²p+14²+1 (mod 6),..., ^p+12^^+1 (mod 6).

• If 0 (mod 6) we have: p+11 (mod 6), ²p+11 (mod 6),...,^p+11 (mod 6).

• If 2 (mod 6) we have: p+13 (mod 6) therefore we cannot have 12+2,-primes if 1and p=6k+1, k1.

• If 4 (mod 6) we have: p+15 (mod 6) ), ²p+15 (mod 6),..., ^p+14^+15 (mod 6).

Finally, we have that ,-primes for p5, 3 can exist only in the cases:

• p5, p=6k-1, =6r, r1;

• p7, p=6k+1, =6r, r1;

• p7, p=6k+1, =6r+4, r0.

The software for determining the ,-primes with =maximum (and limited to primes less than 100000 and 1000) is:

Clear[“Global`*”];

(*,-primes*)

limit=100000;

basepower=1000;

number=2;

maximumnumber=0;

maximumtermsformaximumnumber=0;

maximumpower=0;

For[k=1,klimit,k++,maximumterms=0;

For[=2,basepower,++,

valuetrue=PrimeQ[k];

r=1;

While[valuetrue,valuetrue=valuetrue&&PrimeQ[^r*k+1];r++];

If[maximumterms<r-2,maximumterms=r-2;number=k];

If[maximumtermsformaximumnumber<maximumterms,maximumtermsformaximumnumber=maximumterms;maximumnumber=number;maximumpower=];

];

]

If[maximumtermsformaximumnumber0,Print[“Number=“,maximumnumber];Print[“Base power=“,maximumpower];Print[“Maximum terms=“, maximumtermsformaximumnumber];

For[r=1,r maximumtermsformaximumnumber,r++, Print[maximumpower^r*maximumnumber+1]]]

and we find:

Number=9319

Base power=100

Maximum terms=6

931901

93190001

9319000001

931900000001

93190000000001

9319000000000001

Therefore, the numbers: 9319, 1009319+1, 100²9319+1, 100³9319+1, 100⁴9319+1, 100⁵9319+1, 100⁶9319+1 are all primes.

4 p,-Sophie Germain Sequences

Another type of prime numbers, based also on the idea of Sophie Germain is the following: a sequence will be called a p,-Sophie Germain sequence if p₁=p, p₂=p₁+1=p+1, p₃=p₂+1=²p+ ,..., p_n=p_n-1+1=ⁿp+ are all primes.

Obvious if p2,  must be an even number, otherwise p₂=p₁+1=even.

If p5, p=6k-1 that is p₁=p5 (mod 6) then p₂5+11- (mod 6), p₃5²++11+-² (mod 6) etc.

If =2 then:

• If 0 (mod 6) we have: p_k1 (mod 6) k= .

• If 2 (mod 6) we have: p_k5 (mod 6) k= .

• If 4 (mod 6) we have: p₂3 (mod 6) that is it cannot be prime.

If p7, p=6k+1 that is p₁=p1 (mod 6) then p₂+1 (mod 6), p₃²++1 (mod 6) etc.

If =2 then:

• If 0 (mod 6) we have: p_k1 (mod 6) k= .

• If 2 (mod 6) we have: p₂3 (mod 6) that is it cannot be prime.

• If 4 (mod 6) we have: p₂5 (mod 6), p₃3 (mod 6) that is it cannot be prime.

Finally, we have that p,-primes for p5 can exist only in the cases:

• p5, p=6k-1, =6r, r1;

• p5, p=6k-1, =6r+2, r1;

• p7, p=6k+1, =6r, r1.

The software for determining the p,-Sophie Germain sequence with maximum length (n=maximum) and limited to p10000, 100 is:

Clear[“Global`*”];

(*p,-Sophie Germain sequences*)

limit=10000;

basepower=100;

number=2;

maximumnumber=0;

maximumtermsformaximumnumber=0;

maximumpower=0;

For[k=1,klimit,k++,maximumterms=0;

For[i=2,ibasepower,i++,

valuetrue=PrimeQ[k];

p=1;

While[valuetrue,valuetrue=If[p>1,valuetrue&&PrimeQ[i^p*k+(i^p-1)/(i-1)], valuetrue&&PrimeQ[i*k+1]];p++];

If[maximumterms<p-2,maximumterms=p-2;number=k];

If[maximumtermsformaximumnumber<maximumterms,maximumtermsformaximumnumber=maximumterms;maximumnumber=number;maximumpower=i];

]

]

If[maximumtermsformaximumnumber0,Print[“Number=“,maximumnumber];Print[“Base power=“,maximumpower];Print[“Maximum terms=“,maximumtermsformaximumnumber];For[p=1,pmaximumtermsformaximumnumber, p++,Print[If[p>1,maximumpower^p*maximumnumber+(maximumpower^p-1)/(maximumpower-1),maximumpower*maximumnumber+1]]]]

We find:

Number=37

Base power=48

Maximum terms=5

1777

85297

4094257

196524337

9433168177

that is: p₁=37, p₂=48p₁+1=1777, p₃=48p₂+1=85297, p₄=48p₃+1=4094257,

p₅=48p₄+1=196524337, p₆=48p₅+1=9433168177 are all primes.

5 References

Adler, A. & Coury, J.E. (1995). The Theory of Numbers. London, UK: Jones and Bartlett Publishers International,

Baker, A. (1984). A Concise Introduction to the Theory of Numbers”, Cambridge University Press

Coman, M. (2013). Mathematical Encyclopedia of Integer Classes. Educational Publishers.

Guy, R.K. (1994). Unsolved Problems in Number Theory. 2nd Edition. New York: Springer Verlag.

Hardy, G.H. & Wright, E.M. (1975). Introduction to the Theory of Numbers. 4th Edition. Oxford: Oxford University Press.

Krantz, S.G. (2001). Dictionary of Algebra, Arithmetic and Trigonometry. CRC Press.

Niven, I.; Zuckerman, H.S. & Montgomery, H.L. (1991). An Introduction to the Theory of Numbers. 5th Edition. New York: John Wiley & Sons, Inc.

Sierpinski, W. (1995). Elementary theory of numbers. 2nd Edition. Elsevier.