Numerele perfecte sunt plasate doar in doua spatii din ciurul Teleman,.Locurile acestea sunt evidentiate prin sageti.
Pentru cautarea numerelor perfecte am gasit o alta metoda.
Cu regula lui Euclid ,foarte restrictiva ,se considera doar numerele ''perfect'' de perfecte,unde rolul determinant il au numerele prime nontriviale.
Metoda mea valorifica si numerele prime triviale..
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PROPERTIES OF MERSENNE NUMBERS AND PRIMES
If one looks at the sequence of numbers-
M(p)= 3, 7, 31, 127, 2047 , 8291, 131071, 524287
one notices that its elements are, with the exception of 2047, prime numbers
defined by –
M( p) 2 p 1 with p being the primes 2, 3, 5,7, 11, 13,...
These numbers for any prime p are known as Mersenne Numbers named after the
French cleric and mathematician Marin Mersenne(1588-1648). To date less than
fifty of these numbers have found to be prime. The vast majority of the numbers
M(p) are composite numbers especially at large p. It is our purpose here to
examine these Mersenne Numbers in more detail.
The first thing we notice is that they are always odd numbers when p3 ending in
either 1 or 7 . This means that –
M(p) mod(6) =1
whenever p3 without exception. It implies that Mersenne Numbers lie along the
radial line 6n+1 in the following hexagonal integer spiral diagramThe
M(p) primes are quite sparse compared to the Q Primes along this same radial
line of 6n+1. Only M(p)= 7 and 31 are shown in the diagram. Now 7 and 31 differ
from each other by 24= 6x4. The next gap for M(p) will be 127-
31=96=6x16=12x8=24x4=48x2. These observations suggests that we might
identify Mersenne Numbers by mod values mod(6) followed by mod(12),
mod(24), etc. We have done so and get the following resultsp
M(p) M(p)
mod(6)
M(p)
mod(12)
M(p)
mod(24)
M(p)
Mod(48)
M(p)
Mod(96)
2 3 - - - - -
3 7 1 - - - -
5 31 1 7 7 - -
7 127 1 7 7 31 31
11 2047 1 7 7 31 31
13 8191 1 7 7 31 31
17 131971 1 7 7 31 31
19 524287 1 7 7 31 31
23 8388607 1 7 7 31 31
29 536870911 1 7 7 31 31
31 2147483647 1 7 7 31 31
The table shows that the mod operations using 6, 12 ,24, 48, 96, etc yields the
same constant number for a fixed mod. The mod sequence reads-
1, 7, 31, 127, 511, 2047, 8191, …
where the elements are recognized as–
22m1 1 with m 0, 1, 2, 3, 4, 5,...
For mod(6x2m), the Mersenne Prime will lie along only the single radial line
(6x2n)+const , where the constant represents the appropriate constant for a given
mod. This means, for example , that M(p)=127 will lie along the radial line
48(k)+31 at k=2 in an integer spiral with 48 radial lines. The next Mersenne
Number along this line will be 48( k)+31=2047 with k=42.
In the above table the Merseene Numbers corresponding to p=11, 23, and 29 are
composite, while the others are primes. The mod operations do not distinguish
between the two types of numbers, namely, composite or prime. This distinction
must be found by some type of primality test. The simplest of such tests is that of
Fermat known as Fermat’s Little Theorem. It states that a number is prime if-
p
(a p 1 a)
integer
, where a is a low integer such as 2, 3, 4,… . One can simplify this result by
recasting things using modular arithmetic. Its equivalent form reads-
(ap-1-1)mod(p)=0
So if p=127, we get-
(2126 -1)mod(127)=0
, meaning 127 is a prime. What about the Mersenne Number M(11)=2047? Here
we get-
(32046-1)mod(2047)=1012
indicating that M(11) is a composite. Note that here using a=2 would not work as
a test.
Since in modular arithmetic we have the identity-
(AxBxC) mod(N)=[A mod(N)]x[B mod(N)]x[C mod(N)]
we can simplify the Fermat test for primeness by writing-
(a p1 1)mod(p) [(a( p1) / 2 1)mod(p)]x[(a( p1) / 2 1)mod(p)]
Sometimes this type of breakup can be continued for several more terms making
the mod operations a lot easier. Take the Mersenne Number 8191. To test if it is
prime we write-
[(2 1)mod(8191)] [(2 1)mod(8191)] [0] [2] 0
(2 1)mod(8191)
4095 4095
8190
x x
So it is a Mersenne Prime. To make super-sure, we can replace 2 by any other
positive integer. Doing so, we find-
(38190 1)mod(8191) 0
and the primeness of M(13) is confirmed.
A question often asked is how did Mersenne come up with the form for M(p). The
answer is that he was studying the works of the ancient Greek mathematician
Euclid. In Euclid’s book it is shown that-
1+2=3
1+2+4=7
1+2+4+8=15
1+2+4+8+16=31
1+2+4+8+16+32=63
1+2+4+8+16+32+64=127
or, in general, that-
2 2 1 1
0
N
N
n
n
Now, if N+1=p, then the right hand side of this last expression just represents the
Mersenne Number M(p). It yields the identity-
2 2 1 ( )
1
0
p M p
p
n
n
This is most likely the way Mersenne came up with his number. It says, for
example, that –
M(7)=1+2+4+8+16+32+64=127
It should be pointed out that the Mersenne Numbers are not the only numbers
capable of producing primes via the above approach. By generalizing the above 2n
series, we have-
( 1) 2, 3, 4, 5,... 1
0
where a
N
a a
N N
m
n
If we now take the odd number a=3, then aN+1 will also be odd. Since prime
numbers above p=2 are also odd numbers, it suggests one define a generalized
Mersenne number sequence given by-
F(a,N,b)=aN b. with a 2,3,4,5,..and b 1,2,3,...
In an integral spiral with six radial lines the function F(a,N,b) will lie either along
the line 6N+1 or 6N-1. The corresponding mod operation will yield 1 or 5.
We have played around with various combinations of a and b and find one of the
richest forms for yielding primes is-
F(3,N,2)=K(N)
Here K(N) mod(6)=5 meaning these numbers all lie along the radial line 6N-1 in
the above integral spiral. We have used our PC to find the values of N for which
K(N) is a prime. Here are the results when running over the range 0<N<2000 :
N=1,2,3,4,8.10.14,15,24,26,36,63,98,110,123,126,,139,235,243,315,363,386,391,
494,1131,1220,1503,1858,
This shows a total of 28 primes in the range considered , making the numbers
K(N) more dense in primes than the standard Mersenne Primes of which less than
50 have been found to date. Here is the prime number corresponding to K(1858) :
31858+2=30994973482657325447985147127620089298530431006956594685920
214825663326619113769057695213547484440453837246123010100669127100
955759919127457869223148660164613467627385033860708738672704316650
862181089633382991315337701542754751372995001149340588749380743536
244008880419755202390383627639285895496613970962701489924969795727
825590048078397355734118492863593725501268400567250135296630863031
128376511219475662720257626805673311452744631321737696232779367051
782399514104280263667180701547531032088649276210976147452750560033
145878420041510221647182900875021810238965863623224904713399693521
685908522220999474927503580442427935550946200830198700373449243582
744499189296500580263454833210298365535155326181801482295213278862
293406357314198353241285393759520549296816928599853248072473293309
376537937390628464770980014113878906045022870019937434926857282944
448276114607243265213141322025447691
This 790 digit long prime has, as expected, the value K(1858) mod(6)=5.
We have also played around with other values of ‘a’ and ‘b’. None are found to
be quite as rich in primes as the K(N) numbers.
An additional generalized Mersenne Number sequence which we examined in
some detail is-
F(2,N,1)=2N+1
Searching this function for primes, we were able to find only the five primes
given in the following table-
N F(2,N,1)
1 3
2 5
4 17
8 257
16 65537
In looking at the progressions of N in the table it is clear that they are given by
N=2n. That is, the number F(2,N,1) can also be written as-
F(2,2n,1)=22 1 n
But this function is recognized at once as representing the Fermat Numbers.
Fermat originally thought all these numbers are prime for any positive integer N,
but Euler proved him wrong by showing that 232+1 factors into
4294967297=641 x 6700417
Since that time no additional Fermat Primes have been found, so it is safe to say
that F(2,2n,1) are all composite numbers when n5. Also we have searched
F(2,N,1) over the entire range 16<N<1000 and find no additional primes. It leads
to the conjecture that –
The infinite sequence of numbers F(2,N,1) contains only five primes
corresponding to N=1,2,4,8,16 and no more.
Finally we point out that there are an infinite number of other number sequences
which have a much higher prime number density then the Mersenne Numbers.
One of these which comes to mind is G(N)=N2+(N1). It has a total number of 83
primes out of the first 200 possibilities ( 1<N<100 for both + and – case).
U.H.Kurzweg
Septemver 18, 2015
