- 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 - --- 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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http://mail.yahoo.com - --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>

than p)"

> --- Robert <rw.smith@...> wrote:

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

>

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

> That's equivalent to:

> for every prime p>3, p-2 has at least one prime factor

> and thus is always satisfied.

>

> Phil

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