7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

1/22

PROOFOFANDRICASCONJECTUREANDMINIMUMGAP

BETWEENTWOCONSECUTIVEPRIMES

Ing. Pier Francesco Roggero, Dott. Michele Nardelli, P.A. Francesco Di Noto

Abstract

In this paper, we have described the proof of Andricas Conjecture and Minimum Gap between two

consecutive primes, plus other famous conjectures.

7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

2/22

Versione 1.0

21/2/2013

Pagina 2 di 22

2

Index:

1. PROOFOFANDRICASCONJECTURE ................................................................................................. 32. MINIMUM GAP BETWEEN TWO SUCCESSIVE PRIME NUMBERS SO THAT THERE IS ALWAYS AT

LEAST ONE PRIME NUMBER .................................................................................................................. 83.PROOFOFTHELEGENDRES,OPPERMANNS,CRAMERSANDBROCARDSCONJECTURES......124. CONCLUSION AND SOME MATHEMATICAL CONNECTIONS WITH STRING THEORY.18

7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

3/22

Versione 1.0

21/2/2013

Pagina 3 di 22

3

1. PROOFOFANDRICASCONJECTURE

Andrica's conjecture is a conjecture regarding the gaps between two consecutive prime numbers.

The conjecture states that the inequality

( )nn pp +1 < 1

holds for all , where pn is the nth

prime number. If (pn+1 - pn) denotes the nth

prime gap, then

Andrica's conjecture can also be rewritten as

(1) pn+1 - pn < 2 np + 1

But we can also write

pn+1 - pn = nnnn pppp + ++ 11

Now, because Andrica states that

( )nn pp +1 < 1

we must have at fortiori:

pn+1 - pn < ( )nn pp ++1

This can be seen very simply, if we consider for example the two consecutive primes 127 and 113:

127 113 < ( )113127 +

14 < 21,89.

then necessarily we must have:

7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

4/22

Versione 1.0

21/2/2013

Pagina 4 di 22

4

( )nn pp +1 < 1

( )113127 < 1

0,639 < 1

and then

14 = 21,89*0,639

So, if the Andricas conjecture is true, we must always have:

pn+1 - pn < ( )nn pp ++1

[We note that 0,639 is very near to the aurea section, i.e. 0,61803398 and we note that:

21 * 0,61803398 = 12,9787 13; 21,89 * 0,61803398 = 13,528.

Its interesting to observe that 13 and 21 are Fibonaccis numbers and that, also here, there is a number

very near to the aurea section, number that is very important in various arguments of the string theory]

We see then to prove this inequality, so that it automatically shows also the other of Andrica.

If we assume that the maximum distance between two consecutive prime numbers with Bertrand's

postulate, which is a most extreme case, is then

pn+1 = 2 pn

replacing

pn < ( )nn pp +2

pn < np ( )12 +

np < ( )12 +

np < ( )2

12 +

7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

5/22

Versione 1.0

21/2/2013

Pagina 5 di 22

5

0 np

7/29/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "PROOF OF ANDRICAS CONJECTURE AND MINIMUM

6/22

Versione 1.0

21/2/2013

Pagina 6 di 22

6

In general we assume:

pn+1 = kpn with 1 < k 2

replacing

(k-1) pn < ( )nn pkp +

(k-1) pn < np ( )1+k

np