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## twin primes and 2modp

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• I have been having a mathematical long weekend, first the prime race 1mod3 and 2 mod3 which went nowhere new, and now, 2modp, which will probably also go
Message 1 of 3 , May 1, 2006
I have been having a mathematical long weekend, first the prime race
1mod3 and 2 mod3 which went nowhere new, and now, 2modp, which will
probably also go nowhere. I beg your patience and indulgence a second
time and ask for help.

Can the twin primes conjecture be stated another way?

It seems to me that possible that the following conjecture is equivalent.

"Every prime p, greater than 3, is 2mod(at least one prime less than p)"

It seems to hold for all primes up to prime=250031. I haven't checked
further.

Other than twin primes discussed below, in the range 5-250031 the
prime 239119 gives the highest first 2modp as 2mod437.
The first instance of non twin prime 2modp with highest first 2modp
for p= 3,5,7..97 are
11,37,79,211,223,631,439,853,1249,1459,1741,2503,2281,3433,5779,4663,
5431,4759,7741,16573,8929,8053,12373,13291. This series is not listed
at Sloane's.

I did some fiddling about with the conjecture:

If the conjecture holds then:

1. Take consecutive primes with p0 (>2), p1, and the odd number x = p1+2

2. Then x==2mod(p1)

3. For x prime, p1 is not 0mod(any prime smaller than p1) by
definition of prime numbers

4. Then x is not 2mod(any other prime except p1)

5. An odd number x (not necessarily prime) can be 0mod3, 1mod3 or 2mod3,

6. then the chances of x not being 2mod3 are 2/3 or (3-1)/3

7. The chances of x not being 2modp are (p-1)/p

8. Cumulatively, it appears that the chance, Ca, of x not being
2mod(any prime less than p1) would be given by:

Ca = the product (for primes from p=3 to p=p0) of (p-1)/p

9. If x is prime, then the odds are different, as x cannot be 0mod3,
0modp etc

10. Therefore the chance that x is not 2mod3 is 1/2 is (3-2)/(3-1)

11. chance that x is not 2modp are (p-2)/(p-1)

12. And cumulatively, the chance Cb of a prime x is given by

Cb= the product (for primes from p=3 to p=p(0) of (p-2)/(p-1)

Note: These calculations seem to be heading towards the
Hardy-Littlewood twin prime constant

c2 = product (from p=3 upward) of (p-2)/(p-1)*p/(p-1)

I want to look into this further, but I want to make sure that this
conjecture as stated is not well known before I do so. After
yesterday's debacle concerning prime races, I have gotten shy about
going down well trodden paths.

Regards

Robert Smith
• ... That s equivalent to: for every prime p 3, p-2 has at least one prime factor and thus is always satisfied. Phil () ASCII ribbon campaign ()
Message 2 of 3 , May 1, 2006
--- Robert <rw.smith@...> wrote:
> "Every prime p, greater than 3, is 2mod(at least one prime less than p)"

That's equivalent to:
for every prime p>3, p-2 has at least one prime factor
and thus is always satisfied.

Phil

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• ... than p) ... Phil, as always, to the rescue. Good man! Think I ll stick to prime searching rather than mathematics. I am not yet at class 101 stage, and it
Message 3 of 3 , May 1, 2006
--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- Robert <rw.smith@...> wrote:
> > "Every prime p, greater than 3, is 2mod(at least one prime less
than p)"
>
> That's equivalent to:
> for every prime p>3, p-2 has at least one prime factor
> and thus is always satisfied.
>
> Phil

Phil, as always, to the rescue. Good man!

Think I'll stick to prime searching rather than mathematics. I am not
yet at class 101 stage, and it shows. Still, managed to waste 3 hours
even if non-productively.

For what its worth, the jumping champions (refer to top message) were
as below.

Regards

Robert

11 3
37 5
79 7
211 11
223 13
439 19
853 23
1249 29
1459 31
1741 37
2281 43
3433 47
4663 59
4759 67
7741 71
8053 83
12373 89
12829 101
13591 107
14281 109
17401 127
18211 131
22501 149
25219 151
28201 163
32233 167
32401 179
38011 191
43933 197
47563 199
52963 211
53299 223
53359 229
57601 239
60493 241
68023 251
71191 257
72901 269
77839 277
86269 281
91711 293
95479 307
99223 313
114859 331
129553 353
146833 359
158803 379
159199 397
164011 401
176401 419
194083 421
194479 439
205111 443
227431 467
239023 479
239119 487
253993 499
263071 503
275371 509
284989 521
297589 523
307831 541
321091 547
324901 569
352309 571
356311 587
367069 593
367189 599
378223 613
396733 617
407923 619
416023 643
439471 653
463363 661
497011 701
531343 719
553249 727
571369 743
585163 757
622123 769
650851 787
679843 797
680431 811
680611 821
680623 823
720703 839
756853 863
777901 877
804511 887
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