pier francesco roggero, michele nardelli, francesco di noto - "proof of andrica’s conjecture and...
Post on 03-Apr-2018
219 Views
Preview:
TRANSCRIPT
-
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
top related