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-   -   how is it, primes in the security elements? (https://www.mersenneforum.org/showthread.php?t=23615)

 hal1se 2018-08-29 14:22

how is it, primes in the security elements?

ever prime templates probably twin prime, all template elements very regularly!

3# template probably twin or cousin prime=1
5# template probably twin or cousin prime=3
7# template probably twin or cousin prime=15=3*5
11# tepmlate probably twin or cousin prime=135=3^3*5
13# template probably twin or cousin prime=1485=3^3*5*11
17# template probably twin or cousin prime=22275=3^4*5^2*11
19# template probably twin or cousin prime=378675=3^4*5^2*11*17
23# template probably twin or cousin prime=7952175=3^5*5^2*7*11*17

some one see, any regularly?
you must: look hyper geometric and you must thik complex variables domain!
________________
if we think, only prime template last probably element:
________________

3# to 5# twinprime count, only (6n+5, 6n+6+1) format = 2
(11, 13)
(17, 19)
3#=2*3 to 5#=2*3*5 range
2*3*5 - 2*3 = 2*3*(5-1)=3#*(5-1)
3#*(5-1)/6=(5-1) =4
(3#*n+3#-1, 3#*n+3#+1) format numbers

5# to 7# twinprime count, only (30n+29, 30n+30+1) format = 3
(59, 61)
(149, 151)
(179, 181)
5#=2*3*5 to 7#=2*3*5*7 range
2*3*5*7 - 2*3*5 = 2*3*5*(7-1)=5#*(7-1)
5#*(7-1)/5#=(7-1) =6 times :(5#*n+5#-1, 5#*n+5#+1) format numbers

7# to 11# twinprime count, only (210n+209, 210n+210+1) format = 2
(419, 421)
(1049, 1051)
range:7# to 11#
(7#*n+7#-1, 7#n+7#+1) format numbers= 7#*(11-1)/7#=(11-1)=10

11# to 13# twinprime count, only (2310n+2309, 2310n+2310+1) format = 4
(9239, 9241)
(11549, 11551)
(13679, 13681)
(25409, 25411)
range 11# to 13#
prime template last probably twin format numbers: 13-1=12
13# to 17# twinprime count, only (30030n+30029, 30030n+30030+1) format =

prime template last probably twin format numbers:17-1=16

range:37# to 43#
37# template's last probably twin prime elements:

43-1=42 probaly twin prime numbers.

n=1 to 42, 42 probably twin primes:
(37#*n+37#-1, 37#*n+37#+1)

1*isprime(37#*1+(37#-1))*isprime(37#*1+(37#+1))
2*isprime(37#*2+(37#-1))*isprime(37#*2+(37#+1))
3*isprime(37#*3+(37#-1))*isprime(37#*3+(37#+1))
4*isprime(37#*4+(37#-1))*isprime(37#*4+(37#+1))
5*isprime(37#*5+(37#-1))*isprime(37#*5+(37#+1))
6*isprime(37#*6+(37#-1))*isprime(37#*6+(37#+1))
7*isprime(37#*7+(37#-1))*isprime(37#*7+(37#+1))
8*isprime(37#*8+(37#-1))*isprime(37#*8+(37#+1))
9*isprime(37#*9+(37#-1))*isprime(37#*9+(37#+1))
10*isprime(37#*10+(37#-1))*isprime(37#*10+(37#+1))
11*isprime(37#*11+(37#-1))*isprime(37#*11+(37#+1))
12*isprime(37#*12+(37#-1))*isprime(37#*12+(37#+1))
13*isprime(37#*13+(37#-1))*isprime(37#*13+(37#+1))
14*isprime(37#*14+(37#-1))*isprime(37#*14+(37#+1))
15*isprime(37#*15+(37#-1))*isprime(37#*15+(37#+1))
16*isprime(37#*16+(37#-1))*isprime(37#*16+(37#+1))
17*isprime(37#*17+(37#-1))*isprime(37#*17+(37#+1))
18*isprime(37#*18+(37#-1))*isprime(37#*18+(37#+1))
19*isprime(37#*19+(37#-1))*isprime(37#*19+(37#+1))
20*isprime(37#*20+(37#-1))*isprime(37#*20+(37#+1))
21*isprime(37#*21+(37#-1))*isprime(37#*21+(37#+1))
22*isprime(37#*22+(37#-1))*isprime(37#*22+(37#+1))
23*isprime(37#*23+(37#-1))*isprime(37#*23+(37#+1))
24*isprime(37#*24+(37#-1))*isprime(37#*24+(37#+1))
25*isprime(37#*25+(37#-1))*isprime(37#*25+(37#+1))
26*isprime(37#*26+(37#-1))*isprime(37#*26+(37#+1))
27*isprime(37#*27+(37#-1))*isprime(37#*27+(37#+1))
28*isprime(37#*28+(37#-1))*isprime(37#*28+(37#+1))
29*isprime(37#*29+(37#-1))*isprime(37#*29+(37#+1))
30*isprime(37#*30+(37#-1))*isprime(37#*30+(37#+1))
31*isprime(37#*31+(37#-1))*isprime(37#*31+(37#+1))
32*isprime(37#*32+(37#-1))*isprime(37#*32+(37#+1))
33*isprime(37#*33+(37#-1))*isprime(37#*33+(37#+1))
34*isprime(37#*34+(37#-1))*isprime(37#*34+(37#+1))
35*isprime(37#*35+(37#-1))*isprime(37#*35+(37#+1))
36*isprime(37#*36+(37#-1))*isprime(37#*36+(37#+1))
37*isprime(37#*37+(37#-1))*isprime(37#*37+(37#+1))
38*isprime(37#*38+(37#-1))*isprime(37#*38+(37#+1))
39*isprime(37#*39+(37#-1))*isprime(37#*39+(37#+1))
40*isprime(37#*40+(37#-1))*isprime(37#*40+(37#+1))
41*isprime(37#*41+(37#-1))*isprime(37#*41+(37#+1))
42*isprime(37#*42+(37#-1))*isprime(37#*42+(37#+1))

42 probably prime: n=3 and n=22, only 2 twin prime!
(29682952539239, 29682952539241)
(170676977100629, 170676977100631)

____

range 43# to 47#

43# template's last probably twin prime elementes:

47-1=46 probaly twin prime numbers.

n=1 to 46, 46 probably twin primes:
(43#*n+43#-1, 43#*n+43#+1)

1*isprime(43#*1+(43#-1))*isprime(43#*1+(43#+1))
2*isprime(43#*2+(43#-1))*isprime(43#*2+(43#+1))
3*isprime(43#*3+(43#-1))*isprime(43#*3+(43#+1))
4*isprime(43#*4+(43#-1))*isprime(43#*4+(43#+1))
5*isprime(43#*5+(43#-1))*isprime(43#*5+(43#+1))
6*isprime(43#*6+(43#-1))*isprime(43#*6+(43#+1))
7*isprime(43#*7+(43#-1))*isprime(43#*7+(43#+1))
8*isprime(43#*8+(43#-1))*isprime(43#*8+(43#+1))
9*isprime(43#*9+(43#-1))*isprime(43#*9+(43#+1))
10*isprime(43#*10+(43#-1))*isprime(43#*10+(43#+1))
11*isprime(43#*11+(43#-1))*isprime(43#*11+(43#+1))
12*isprime(43#*12+(43#-1))*isprime(43#*12+(43#+1))
13*isprime(43#*13+(43#-1))*isprime(43#*13+(43#+1))
14*isprime(43#*14+(43#-1))*isprime(43#*14+(43#+1))
15*isprime(43#*15+(43#-1))*isprime(43#*15+(43#+1))
16*isprime(43#*16+(43#-1))*isprime(43#*16+(43#+1))
17*isprime(43#*17+(43#-1))*isprime(43#*17+(43#+1))
18*isprime(43#*18+(43#-1))*isprime(43#*18+(43#+1))
19*isprime(43#*19+(43#-1))*isprime(43#*19+(43#+1))
20*isprime(43#*20+(43#-1))*isprime(43#*20+(43#+1))
21*isprime(43#*21+(43#-1))*isprime(43#*21+(43#+1))
22*isprime(43#*22+(43#-1))*isprime(43#*22+(43#+1))
23*isprime(43#*23+(43#-1))*isprime(43#*23+(43#+1))
24*isprime(43#*24+(43#-1))*isprime(43#*24+(43#+1))
25*isprime(43#*25+(43#-1))*isprime(43#*25+(43#+1))
26*isprime(43#*26+(43#-1))*isprime(43#*26+(43#+1))
27*isprime(43#*27+(43#-1))*isprime(43#*27+(43#+1))
28*isprime(43#*28+(43#-1))*isprime(43#*28+(43#+1))
29*isprime(43#*29+(43#-1))*isprime(43#*29+(43#+1))
30*isprime(43#*30+(43#-1))*isprime(43#*30+(43#+1))
31*isprime(43#*31+(43#-1))*isprime(43#*31+(43#+1))
32*isprime(43#*32+(43#-1))*isprime(43#*32+(43#+1))
33*isprime(43#*33+(43#-1))*isprime(43#*33+(43#+1))
34*isprime(43#*34+(43#-1))*isprime(43#*34+(43#+1))
35*isprime(43#*35+(43#-1))*isprime(43#*35+(43#+1))
36*isprime(43#*36+(43#-1))*isprime(43#*36+(43#+1))
37*isprime(43#*37+(43#-1))*isprime(43#*37+(43#+1))
38*isprime(43#*38+(43#-1))*isprime(43#*38+(43#+1))
39*isprime(43#*39+(43#-1))*isprime(43#*39+(43#+1))
40*isprime(43#*40+(43#-1))*isprime(43#*40+(43#+1))
41*isprime(43#*41+(43#-1))*isprime(43#*41+(43#+1))
42*isprime(43#*42+(43#-1))*isprime(43#*42+(43#+1))
43*isprime(43#*43+(43#-1))*isprime(43#*43+(43#+1))
44*isprime(43#*44+(43#-1))*isprime(43#*44+(43#+1))
45*isprime(43#*45+(43#-1))*isprime(43#*45+(43#+1))
46*isprime(43#*46+(43#-1))*isprime(43#*46+(43#+1))

46 probably prime: n=23 , only 1 twin prime!
( (43#*23+(43#-1)) , (43#*23+(43#+1)) )
(313986271960080719, 313986271960080721)

_________
range: 47# to 53#

47# template's last probably twin prime elements:

53-1=52 probaly twin prime numbers.

n=1 to 52, 52 probably twin primes:
(47#*n+47#-1, 47#*n+47#+1)

1*isprime(47#*1+(47#-1))*isprime(47#*1+(47#+1))
2*isprime(47#*2+(47#-1))*isprime(47#*2+(47#+1))
3*isprime(47#*3+(47#-1))*isprime(47#*3+(47#+1))
4*isprime(47#*4+(47#-1))*isprime(47#*4+(47#+1))
5*isprime(47#*5+(47#-1))*isprime(47#*5+(47#+1))
6*isprime(47#*6+(47#-1))*isprime(47#*6+(47#+1))
7*isprime(47#*7+(47#-1))*isprime(47#*7+(47#+1))
8*isprime(47#*8+(47#-1))*isprime(47#*8+(47#+1))
9*isprime(47#*9+(47#-1))*isprime(47#*9+(47#+1))
10*isprime(47#*10+(47#-1))*isprime(47#*10+(47#+1))
11*isprime(47#*11+(47#-1))*isprime(47#*11+(47#+1))
12*isprime(47#*12+(47#-1))*isprime(47#*12+(47#+1))
13*isprime(47#*13+(47#-1))*isprime(47#*13+(47#+1))
14*isprime(47#*14+(47#-1))*isprime(47#*14+(47#+1))
15*isprime(47#*15+(47#-1))*isprime(47#*15+(47#+1))
16*isprime(47#*16+(47#-1))*isprime(47#*16+(47#+1))
17*isprime(47#*17+(47#-1))*isprime(47#*17+(47#+1))
18*isprime(47#*18+(47#-1))*isprime(47#*18+(47#+1))
19*isprime(47#*19+(47#-1))*isprime(47#*19+(47#+1))
20*isprime(47#*20+(47#-1))*isprime(47#*20+(47#+1))
21*isprime(47#*21+(47#-1))*isprime(47#*21+(47#+1))
22*isprime(47#*22+(47#-1))*isprime(47#*22+(47#+1))
23*isprime(47#*23+(47#-1))*isprime(47#*23+(47#+1))
24*isprime(47#*24+(47#-1))*isprime(47#*24+(47#+1))
25*isprime(47#*25+(47#-1))*isprime(47#*25+(47#+1))
26*isprime(47#*26+(47#-1))*isprime(47#*26+(47#+1))
27*isprime(47#*27+(47#-1))*isprime(47#*27+(47#+1))
28*isprime(47#*28+(47#-1))*isprime(47#*28+(47#+1))
29*isprime(47#*29+(47#-1))*isprime(47#*29+(47#+1))
30*isprime(47#*30+(47#-1))*isprime(47#*30+(47#+1))
31*isprime(47#*31+(47#-1))*isprime(47#*31+(47#+1))
32*isprime(47#*32+(47#-1))*isprime(47#*32+(47#+1))
33*isprime(47#*33+(47#-1))*isprime(47#*33+(47#+1))
34*isprime(47#*34+(47#-1))*isprime(47#*34+(47#+1))
35*isprime(47#*35+(47#-1))*isprime(47#*35+(47#+1))
36*isprime(47#*36+(47#-1))*isprime(47#*36+(47#+1))
37*isprime(47#*37+(47#-1))*isprime(47#*37+(47#+1))
38*isprime(47#*38+(47#-1))*isprime(47#*38+(47#+1))
39*isprime(47#*39+(47#-1))*isprime(47#*39+(47#+1))
40*isprime(47#*40+(47#-1))*isprime(47#*40+(47#+1))
41*isprime(47#*41+(47#-1))*isprime(47#*41+(47#+1))
42*isprime(47#*42+(47#-1))*isprime(47#*42+(47#+1))
43*isprime(47#*43+(47#-1))*isprime(47#*43+(47#+1))
44*isprime(47#*44+(47#-1))*isprime(47#*44+(47#+1))
45*isprime(47#*45+(47#-1))*isprime(47#*45+(47#+1))
46*isprime(47#*46+(47#-1))*isprime(47#*46+(47#+1))
47*isprime(47#*47+(47#-1))*isprime(47#*47+(47#+1))
48*isprime(47#*48+(47#-1))*isprime(47#*48+(47#+1))
49*isprime(47#*49+(47#-1))*isprime(47#*49+(47#+1))
50*isprime(47#*50+(47#-1))*isprime(47#*50+(47#+1))
51*isprime(47#*51+(47#-1))*isprime(47#*51+(47#+1))
52*isprime(47#*52+(47#-1))*isprime(47#*52+(47#+1))

52 probably twin prime, but only 1 twin prime:
for n=36:
(22750921955774182169, 22750921955774182171)

_____________
range: 53# to 59#, last prime templates probably twin primes:
58 probably twin prime.

1*isprime(53#*1+(53#-1))*isprime(53#*1+(53#+1))
2*isprime(53#*2+(53#-1))*isprime(53#*2+(53#+1))
3*isprime(53#*3+(53#-1))*isprime(53#*3+(53#+1))
4*isprime(53#*4+(53#-1))*isprime(53#*4+(53#+1))
5*isprime(53#*5+(53#-1))*isprime(53#*5+(53#+1))
6*isprime(53#*6+(53#-1))*isprime(53#*6+(53#+1))
7*isprime(53#*7+(53#-1))*isprime(53#*7+(53#+1))
8*isprime(53#*8+(53#-1))*isprime(53#*8+(53#+1))
9*isprime(53#*9+(53#-1))*isprime(53#*9+(53#+1))
10*isprime(53#*10+(53#-1))*isprime(53#*10+(53#+1))
11*isprime(53#*11+(53#-1))*isprime(53#*11+(53#+1))
12*isprime(53#*12+(53#-1))*isprime(53#*12+(53#+1))
13*isprime(53#*13+(53#-1))*isprime(53#*13+(53#+1))
14*isprime(53#*14+(53#-1))*isprime(53#*14+(53#+1))
15*isprime(53#*15+(53#-1))*isprime(53#*15+(53#+1))
16*isprime(53#*16+(53#-1))*isprime(53#*16+(53#+1))
17*isprime(53#*17+(53#-1))*isprime(53#*17+(53#+1))
18*isprime(53#*18+(53#-1))*isprime(53#*18+(53#+1))
19*isprime(53#*19+(53#-1))*isprime(53#*19+(53#+1))
20*isprime(53#*20+(53#-1))*isprime(53#*20+(53#+1))
21*isprime(53#*21+(53#-1))*isprime(53#*21+(53#+1))
22*isprime(53#*22+(53#-1))*isprime(53#*22+(53#+1))
23*isprime(53#*23+(53#-1))*isprime(53#*23+(53#+1))
24*isprime(53#*24+(53#-1))*isprime(53#*24+(53#+1))
25*isprime(53#*25+(53#-1))*isprime(53#*25+(53#+1))
26*isprime(53#*26+(53#-1))*isprime(53#*26+(53#+1))
27*isprime(53#*27+(53#-1))*isprime(53#*27+(53#+1))
28*isprime(53#*28+(53#-1))*isprime(53#*28+(53#+1))
29*isprime(53#*29+(53#-1))*isprime(53#*29+(53#+1))
30*isprime(53#*30+(53#-1))*isprime(53#*30+(53#+1))
31*isprime(53#*31+(53#-1))*isprime(53#*31+(53#+1))
32*isprime(53#*32+(53#-1))*isprime(53#*32+(53#+1))
33*isprime(53#*33+(53#-1))*isprime(53#*33+(53#+1))
34*isprime(53#*34+(53#-1))*isprime(53#*34+(53#+1))
35*isprime(53#*35+(53#-1))*isprime(53#*35+(53#+1))
36*isprime(53#*36+(53#-1))*isprime(53#*36+(53#+1))
37*isprime(53#*37+(53#-1))*isprime(53#*37+(53#+1))
38*isprime(53#*38+(53#-1))*isprime(53#*38+(53#+1))
39*isprime(53#*39+(53#-1))*isprime(53#*39+(53#+1))
40*isprime(53#*40+(53#-1))*isprime(53#*40+(53#+1))
41*isprime(53#*41+(53#-1))*isprime(53#*41+(53#+1))
42*isprime(53#*42+(53#-1))*isprime(53#*42+(53#+1))
43*isprime(53#*43+(53#-1))*isprime(53#*43+(53#+1))
44*isprime(53#*44+(53#-1))*isprime(53#*44+(53#+1))
45*isprime(53#*45+(53#-1))*isprime(53#*45+(53#+1))
46*isprime(53#*46+(53#-1))*isprime(53#*46+(53#+1))
47*isprime(53#*47+(53#-1))*isprime(53#*47+(53#+1))
48*isprime(53#*48+(53#-1))*isprime(53#*48+(53#+1))
49*isprime(53#*49+(53#-1))*isprime(53#*49+(53#+1))
50*isprime(53#*50+(53#-1))*isprime(53#*50+(53#+1))
51*isprime(53#*51+(53#-1))*isprime(53#*51+(53#+1))
52*isprime(53#*52+(53#-1))*isprime(53#*52+(53#+1))
53*isprime(53#*53+(53#-1))*isprime(53#*53+(53#+1))
54*isprime(53#*54+(53#-1))*isprime(53#*54+(53#+1))
55*isprime(53#*55+(53#-1))*isprime(53#*55+(53#+1))
56*isprime(53#*56+(53#-1))*isprime(53#*56+(53#+1))

56 probaly prime, but: n=27, only 1 twinprime:
(912496437361321252439, 912496437361321252441)
__________
middle point of range:
(53# + 57#) / 2 = 53# * (1 + 57)/2 = 53# * 58/2 = 53# * 29

ln (middle point of range)=ln(945085595838511297170)
=48,297807174891551584668165091634

this 53# to 59# range every ((48,2978)^2 *(3/4)=1749,5 numbers: 1 twin prime (average!)

56 probably twin, and every 1749,5 numbers average 1 twin.

probabaly results:
range twin count=0 highly reality.
range twin count=1 low reality.
range twin count=2 less low reality.
...
very rare result:
range twin count=56 very low reality.

but range twin count= 0,1,2,3 highly reality.

if even, many ranges prime templates last probably twin prime range reel count=0,
but some ranges at least 1 twin prime reel count posible reality.

don't forget: twin prime real count at least 1, posible reality > 0, so: infinity twin prime there are, very simple!
_______
range: 1009# to 1013#
prime template 1009#, last element probably twin prime numbers:
1013-1=1012 times!
middle point of range:
(1009# + 1013#)/2=1009#(1+1013)/2=
=1009#*507

ln(1009# *507)=969,390
every (3/4)*(969,390)^2= 704787,7 numbers, 1 twinprime average range count!

probably twin prime:1012
every 704788 numbers 1 twin prime, average!

so: same think!
probabaly results:
range last element twin count=0 very highly reality.
range last elelement twin count=1:this is low reality 1012/704788=1/696,4 reality
range twin count=2 more less low reality.
...
very rare result:
range twin count=1012 very low reality.

but range twin count= 0,1,2,3 highly reality.

if this range twin count=0, other some big ranges at least 1 twinprime count.

so infinity twin primes are there: because low reality result=1,2,3,... low but posible!
low reality, but reality >0
so every prime templates last element, primoryel ranges real twin prime count infinity!
very simple!

test: 1009# template only last element probably twin prime control:

1*isprime(1009#*1+(1009#-1))*isprime(1009#*1+(1009#+1))
2*isprime(1009#*2+(1009#-1))*isprime(1009#*2+(1009#+1))
3*isprime(1009#*3+(1009#-1))*isprime(1009#*3+(1009#+1))
4*isprime(1009#*4+(1009#-1))*isprime(1009#*4+(1009#+1))
5*isprime(1009#*5+(1009#-1))*isprime(1009#*5+(1009#+1))
6*isprime(1009#*6+(1009#-1))*isprime(1009#*6+(1009#+1))
7*isprime(1009#*7+(1009#-1))*isprime(1009#*7+(1009#+1))
8*isprime(1009#*8+(1009#-1))*isprime(1009#*8+(1009#+1))
9*isprime(1009#*9+(1009#-1))*isprime(1009#*9+(1009#+1))
10*isprime(1009#*10+(1009#-1))*isprime(1009#*10+(1009#+1))
...

1011*isprime(1009#*1011+(1009#-1))*isprime(1009#*1011+(1009#+1))
1012*isprime(1009#*1012+(1009#-1))*isprime(1009#*1012+(1009#+1))

range 1009# to 1013# only a few hundreds decimal digits (about 500)

1012 twinprime, so 2024 prime test only < 1 minutes.
result: no twin prime count, in the 1012 probably twins.
very normal!
because:this is low reality 1012/704788=1/696,4 reality!
but think:
infinty real twin prime in: prime template last element probably twin elements!

because :
count 1 or 2 or 3, posibilities > 0 then,
some primoryel big ranges twin prime count must be at least 1 or greater than 1.
so:infinty twin prime there are, very simple!

_________________
if numbers < 1e5 decimal digits, isprime function very fast or only fast!
if 1e5 decimal digits < numbers < 1e8 decimal digits,
isprime function medium fast, may be need many parellel cores!
but numbers > 1e10 decimal digits isprime function very slow and very bad!
may be, somone(s) combined puzzle, near future:
numbers =~ 1e16 decimal digits, only a few minute prime test!
this method posible, don't forget!

how is it, primes in the security elements?
prime numbers not safe!
people are crude idiots :(

 hal1se 2018-08-29 19:02

find easy twin prime, in the infinty twin primes

 Batalov 2018-08-30 02:06

[QUOTE=hal1se;494866]
people are crude idiots :([/QUOTE]