pier francesco roggero, michele nardelli, francesco di noto - "the new mersenne prime numbers"

7/27/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "The New Mersenne Prime Numbers"

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THE NEW MERSENNE PRIME NUMBERS

Pier Francesco Roggero, Michele Nardelli1,2

, Francesco Di Noto

1 Dipartimento di Scienze della Terra

Universit degli Studi di Napoli Federico II, Largo S. Marcellino, 10

80138 Napoli, Italy

2 Dipartimento di Matematica ed Applicazioni R. Caccioppoli

Universit degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie

Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

In this paper we examine in detail the new Mersenne Prime Numbers with their

important connections with Fermats prime numbers. Furthermore, we have

described some mathematical connections between some equations regarding theprimitive divisors of Mersenne numbers and some equations concerning the String

Theory


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Index:

1. THE NEW MERSENNE PRIME NUMBERS.......................................................................................... 32. ANALYSIS ON NEW MERSENNE NUMBERS...................................................................................... 83. CONNECTION NNM WITH FERMAT NUMBERS................................................................................13

4. ON SOME EQUATIONS CONCERNING THE PRIMITIVE DIVISORS OF MERSENNE NUMBERS20

5. REFERENCES. 26


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1. THE NEW MERSENNE PRIME NUMBERS

In this paper we will discuss the New Mersenne numbers NNM, of form 2n

+1, of

their statistical distribution, similar to that of the old Mersenne numbers (of the

form 2n-1, prime and not prime)). but, as we will see in the end, there are only four

prime numbers in numbers NNM, based on the same considerations arithmetic

(common factors) to only five prime numbers of Fermat.

While the classical Mersenne numbers, Mp are related to perfect numbers, theNNM are related to Fermat prime numbers, and there are in finite number only 4,

while the Fermat primes are only 5 known.

They are complementary to the first, but with similar numerically and similar

statistical distribution.

While for the known Mersenne numbers the exponent of 2 is odd, and therefore

potentially prime (the thing that more interested) in our NNM instead is even, andthen to get prime numbers we must add 1 instead of subtracting:

2

n

1

for the Mersenne numbers, with n odd and prime, and with form 6k +1, for the

reasons at the time shown in previous work.

2n

+ 1

for the NNM, with n even, and then composite, and with form 6k -1.

We make a small table with the first ten numbers NNM. If they will be of interest

from the mathematical point of view, we will return later on this topic, here only

introduced to the attention of mathematicians.


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TABLE

n even 2n

+1 Prime or

not

Form 6k-1 Ratio Exponent /

number of digits

in NNM

2 5 si Si k=1 24 17 si Si k=3 4/2=2

6 65 = 5*13 no Si k=10 6/2=38 257 si Si k= 43 8/3= 2,66

10 1 025 =52*41 no Si k=171 10/4 = 2,5

12 4 097=17*241 no Si k= 683 12/4=3

14 16385=5*29*113 no Si k=2731 14/5=2,8

16 65 537 si Si k=10923 16/5= 3,2

18 262145=5*13*37*10

9

no Si k=43691 18/6=3

20 1 048 577=17*61681 no Si k=174763 20/7= 2,85

As we can see, the next ratio n / number of digits in NNM grows slowly, in the end,

after a few dozen other ratio values will reach almost constant 3.321 as for thenumbers of Mersenne classics, as we shall see in subsequent statistical tables.

We also note that among the first 10 values of NNM there are four primes (and

only, as we shall see at the end), while for the Mersenne numbers classics there are

five, but taking into account here that the exponent is highest (31), while for the

NNM is 20, and this justifies the units more.

Now, we note that the number 8, is connected with the modes that correspond tothe physical vibrations of a superstring by the following Ramanujan function:


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( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

.

Furthermore, the number that result from 8/3 = 2,666666667 is equal to

2,66666667 that is a frequency of the universal musical system based on Phi (Phi =

aurea ratio = 1,61803398875

But if we consider as maximum exponent 20 for a comparison, then we havenumbers early in equal measure, 4 and 4 for both versions. So the statistical

distribution is almost similar, as expected.

But the formula for the NNM cannot be helpful in finding prime numbers more

large, using exponents equal to 2, instead of the prime odd and as for the first

version, since the prime numbers in NNM are only four, because of the connectionswith Fermat primes.

About even perfect numbers, however, cannot be connected to the NNM, we report

the proof of the form 6k - 2 perfect numbers in this case, but similar to the one

with the formula of perfect numbers):

for the odd powers of 2 are in the form 6k - 4, and the even powers are in the form

6k - 2, we have (6k - 4) - (6k '- 2) = 6k - 4 + 6k '+ 2 = 6k - 6k' - 4 - 2 = 6 (k - k ') - 4 +2 = 6k'' - 2.

Example:

32 4 = 28 = 30 - 2

6*6 - 4 (6*1 2) = 36 - 4 (6*1 - 2) = 36 4 6 + 2 = 6 (6 - 1) - 2 =


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6*5 - 2 = 30 - 2 =28

Without this clarification about the even perfect numbers of the form 6k 2 (the

odd perfect numbers do not exist), we now turn to the findings of this Chapter.


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CONCLUSIONS

This is only a proposal for a extension of the concept of Mersenne numbers to

numerical form:

2n

+1

possibly to be explored in the future if interested by mathematicians professionalor amateur, and observations of any possible mathematical applications, however,

excluded by the above (existence of only 4 primes in NNM) attempting to find large

prime numbers similar to how you do it for the old Mersenne primes,

As is well known, the 48th number, with about 17 million digits recently

discovered.

But this could very well provide our Universal Rule for find all prime numbers

(Ref.1 and Ref.2, in English, and Ref.3), without having to resort to the formulas ofMersenne, which would require more computation time, especially for primes with

thousands or millions of digits, useful for cryptographic purposes (RSA in

particular, while the ECC encryption offers the same security with Prime

Numbers length ten times less than the RSA numbers and much more difficult to

crack by hackers).


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2. ANALYSIS ON NEW MERSENNE NUMBERS

Let us examine in detail the new Mersenne numbers NNM and see what give origin

to prime numbers.

NNM = 2

n

+ 1

with n even (2, 4, 6, )

We have the following table

TAB 1:

n 2n+1 factorization

2 54 17

6 65

8 257

10 1025

12 4097 17 241

14 16385

16 65537

18 262145

20 1048577 17 6168122 4194305

24 16777217 97 257 673

26 67108865

28 268435457 17 15790321

30 1073741825

32 4294967297 641 6700417

34 17179869185

36 68719476737 17 241 433 38737


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38 27487790694540 1099511627777 257 4278255361

42 4398046511105

44 17592186044417 17 353 2931

46 70368744177665

48 281474976710657 193 65537 22253377

All numbers NNM ensuing have as the last digit 5 or 7.

Consequently, only those with last digit 7 can give prime numbers.

So logically we need to consider all n that are only multiples of 4 and we have:

NNM = 2n

+ 1

with n multiple of 4 (4, 8, 12, ) with 8 connected with the modes thatcorrespond to the physical vibrations of a superstring by the following Ramanujanfunction:

( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

.


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Thus we have the new table:

TAB. 2:


4 17

8 257

12 4097 17 24116 65537

20 1048577 17 61681

24 16777217 97 257 673

28 268435457 17 15790321

32 4294967297 641 6700417

36 68719476737 17 241 433 38737

40 1099511627777 257 4278255361

44 17592186044417 17 353 2931

48 281474976710657 193 65537 22253377

Considering only the n multiples of 4 in the first 12 numbers we have 4 prime

numbers (including 5 that derives from p = 2).

In the "old" Mersenne numbers Mp of the form:

Mp = 2p

1

with p prime, all the prime numbers that emerge from it have a final digit 1 or 7,except for the initial 3 due to the fact that p = 2 even number but prime number.

In this case, for p 61 (roughly as n 48), there are 9 primes, twice before and

considering that the latest digits are 1 or 7 instead of only 7 the bill back.


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In addition, each

n = 8k +4 with k = 0, 1, 2, is divisible by 17

n = 24k + 12 with k = 0, 1, 2, is divisible by 241

plus other factors as shown by the TAB .2

All of the factors resulting lower the probability of finding new Mersenne primes.

We can therefore say:

They generate about half of prime numbers with NNM than older Mersenne

numbers Mp.

We note that in 8k + 4 and 24k + 12 there are 8 and 24 where 8 is connected with

the modes that correspond to the physical vibrations of a superstring by the

following Ramanujan function:

( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

Furthermore, with regard the number 24, it is related to the physical vibrations of

the bosonic strings by the following Ramanujan function:


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( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

The NNM are interesting from a theoretical point of view as they complete the

symmetry arithmetic 6k -1 and 6k +1 as general forms of prime numbers.

At first glance it would seem that there should be no difference between the

numbers Mp and NNM to a certain n, but in reality, with this work, we found that

this is not so because the numbers are already numerous Mp (recently it was

discovered the 48th, with about 17 million digits), while the prime numbers inNNM are only four, see Paragraph 3.


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3. CONNECTION NNM WITH FERMAT NUMBERS

Returning to Table 2 we try to extend it with the factorization extended to all

numbers according to the trickery that we already know a priori what are the

factors of numbers NNM:

TAB. 3:


4 17 17

8 257 257

12 4097 17 241

16 65537 65537

20 1048577 17 61681

24 16777217 97 257 673

28 268435457 17 15790321

32 4294967297 641 6700417

36 68719476737 17 241 433 38737

40 1099511627777 257 4278255361

44 17592186044417 17 353 2931

48 281474976710657 193 65537 22253377

52 4,50359963E+15 17 858001 308761441

56 7,20575940E+16 257 5153 54410972897

60 1,15292150E+18 17 241 61681 4562284561

64 1,84467441E+19 274177 67280421310721

68 2,95147905E+20 17^2 354689 2879347902817

72 4,72236648E+21 97 257 577 673 487824887233

76 7,55578637E+22 17 1217 148961 24517014940753

80 1,20892582E+24 65537 414721 44479210368001

84 1,93428131E+25 17 241 3361 15790322 89959882481

88 3,09485010E+26 257 229153 119782433 43872038849

92 4,95176016E+27 17

96 7,92281625E+28 641 6700417 18446744069414584321

100 1,26765060E+30 17


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104 2,02824096E+31 257

108 3,24518554E+32 17 241

112 5,19229686E+33 65537

116 8,30767497E+34 17

120 1,32922800E+36 257

124 2,12676479E+37 17

128 3,40282367E+38 59649589127497217 5704689200685129054721

132 5,44451787E+39 17 241

136 8,71122859E+40 257

140 1,39379657E+42 17

144 2,23007452E+43 65537

148 3,56811923E+44 17

152 5,70899077E+45 257

156 9,13438523E+46 17 241

160 1,46150164E+48 641 3602561 6700417 94455684953484563055991838558081

164 2,33840262E+49 17

168 3,74144419E+50 257

172 5,98631071E+51 17

176 9,57809713E+52 65537

180 1,53249554E+54 17 241

184 2,45199287E+55 257

188 3,92318858E+56 17

192 6,27710174E+57 769 274177 67280421310721 442499826945303593556473164314770689

196 1,00433628E+59 17

200 1,60693804E+60 257

204 2,57110087E+61 17 241

208 4,11376139E+62 65537

212 6,58201823E+63 17

216 1,05312292E+65 257

220 1,68499667E+66 17

224 2,69599467E+67 641 6700417

228 4,31359147E+68 17 241

232 6,90174635E+69 257

236 1,10427942E+71 17

240 1,76684706E+72 65537

244 2,82695530E+73 17

248 4,52312849E+74 257


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252 7,23700558E+75 17 241

256 1,15792089E+77 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

260 1,85267343E+78 17

264 2,96427748E+79 257

268 4,74284398E+80 17

272 7,58855036E+81 65537

276 1,21416806E+83 17 241

280 1,94266889E+84 257

284 3,10827023E+85 17

288 4,97323236E+86 641 6700417

292 7,95717178E+87 17

296 1,27314749E+89 257

300 2,03703598E+90 17 241

304 3,25925756E+91 65537

308 5,21481210E+92 17

312 8,34369936E+93 257

316 1,33499190E+95 17

320 2,13598704E+96 274177 67280421310721

324 3,41757926E+97 17 241

328 5,46812681E+98 257

332 8,74900290E+99 17

336 1,39984046E+101 65537

340 2,23974474E+102 17

344 3,58359159E+103 257

348 5,73374654E+104 17 241

352 9,17399446E+105 641 6700417

356 1,46783911E+107 17

360 2,34854258E+108 257

364 3,75766813E+109 17

368 6,01226901E+110 65537

372 9,61963042E+111 17 241

376 1,53914087E+113 257

380 2,46262539E+114 17

384 3,94020062E+115 59649589127497217 5704689200685129054721

388 6,30432099E+116 17

392 1,00869136E+118 257

396 1,61390617E+119 17 241

400 2,58224988E+120 65537


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404 4,13159980E+121 17

408 6,61055969E+122 257

412 1,05768955E+124 17

416 1,69230328E+125 641 6700417

420 2,70768525E+126 17 241

424 4,33229640E+127 257

428 6,93167424E+128 17

432 1,10906788E+130 65537

436 1,77450860E+131 17

440 2,83921377E+132 257

444 4,54274203E+133 17 241

448 7,26838724E+134 274177 67280421310721

452 1,16294196E+136 17

456 1,86070713E+137 257

460 2,97713141E+138 17

464 4,76341026E+139 65537

468 7,62145642E+140 17 241

472 1,21943303E+142 257

476 1,95109284E+143 17

480 3,12174855E+144 641 6700417

484 4,99479768E+145 17

488 7,99167629E+146 257

492 1,27866821E+148 17 241

496 2,04586913E+149 65537

500 3,27339061E+150 17

504 5,23742497E+151 257

508 8,37987996E+152 17

512 1,34078079E+1542424833, 7455602825647884208337395736200454918783366342657,

741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737


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In fact, the prime factors of NNM are known according to this table of the

exponents n

TAB. 4:

17 n=8k+4

257 n=16k+8

65537 n=32k+16

641, 6700417 n=64k+32

274177, 67280421310721 n=128k+64

59649589127497217 5704689200685129054721 n=256k+128

1238926361552897, 93461639715357977769163558199606896584051237541638188580280321 n=512k+256

2424833, 7455602825647884208337395736200454918783366342657,

741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737n=1024k+512


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Now we have the connection with Fermat numbers:

122 +=n

Fn

We know that there are only 5 Fermat primes (see our proof in Ref 6).

Now from the TAB. 4 we have "overlapping" total of ALL the exponents n

multiples of 4 and ALL its factors are ALL the factors of Fermat numbers.

In fact all the exponents n, in multiples of 4, are given by the following table:

n=8k+4

n=16k+8

n=32k+16

n=64k+32

n=128k+64

n=256k+128

n=512k+256

n=1024k+512

In this way we have:

n = 4, 8 , 12, 16 . ALL multiples of 4

Also here we note that 8, 16, 32, 64, 128, 256, 512 and 1024 are all divisible for 8,

i.e. the number that isconnected with the modes that correspond to the physical

vibrations of a superstring by the following Ramanujan function


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( )

++

+

=

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

.

All prime factors of both numbers NNM and Fermat numbers are exactly the same

because the frequencies of n are only due ton22 .

This result validates and demonstrates that there are only 5 Fermat primes.

In addition:

1)The numbers of Fermat numbers are a subset of NNM.2)This demonstrates that also for numbers NNM we only have 4 prime

numbers, the last number is the prime number 65537 and that there are

others for both the group NNM and the group of Fermat.

CONCLUSIONS

1) and 2) are the most interesting and unexpected results of this work, born from

the idea, now proved incorrect, we can find other very large prime numberssimilar size as the old Mersenne primes Mp.

Instead we have found only 4, with mathematical certainty that there are no

others.


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4. ON SOME EQUATIONS CONCERNIG PRIMITIVE DIVISORS OF

MERSENNE NUMBERS

In his interesting paper On the sum

12

1

nd

d (1971), the mathematician Paul

Erdos considered the divisor function ( ) =md

dm /11 restricted to Mersenne

numbers; that is, numbers m of the form 2

n

1. It is well known that

( ) ( )mOm loglog1 = . (1)

Thus letting 12 = nm , we trivially have

( ) ( )nOn log121 = . (2)

THEOREM 1

There is a set of natural numbers S of logarithmic density 1 such that

( ) 0lim,

=

nnEnSn

. (3)

We know that for each 0> the set ( )T of n with ( ) >nnE has logarithmic density

0. For each prime q , let

P ( ) {pq = prime: ,mod1 qp 2 is not a qth power }pmod .

We have that:

( )( )( )( )

=

plqxp qPpxp

q

q xOp

p

p

p

|, ,

/11log

11. (4)

And by (4) follow also that


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( )( )( )

=

=

=ndmdl

m dnA,

/1 (17)

so that

( ) ( )=nm

m nAnA . (18)

There is an 0x such that for all 0xx and any m ,

# ( ){ }

+=

x

xxxmdlxd

loglog2

logloglog3logexp: . (19)


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Thus by partial summation

( )( )

=

pier francesco roggero, michele nardelli, francesco di noto - "the new mersenne prime numbers"

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