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Re: A bit more on Goldbach's and Primes
Jim Choate wrote:
>
>
>
> Goldbach's Conjecture:
>
> Any even number >2 can be represented as the sum of two prime numbers.
>
>
> Goldbach's Extension:
>
> Any even number >=6 can be represented as the sum of three even numbers. 2
> plus two evens that are each 1 less than a prime.
Jim,
The "extension" is just a trivial consequence of the original conjecture.
Proof: Take any even number N >= 6.
According to the original conjecture, there are two prime numbers P1 and
P2 such that N=p1+p2.
Rewriting this: N = 2 + (p1-1) + (p2-1).
That is, two plus two evens that are each one less than a prime.
That's eighth grade math.
>
> Process:
>
> 1. Pick an even number
> 2. Select largest prime that is smaller by at least 4.
> 3. Add 1.
> 4. The difference between 1 and 3 should be a p-1. Add one to the
> difference and it must be in the list of primes. If it's not then
if it is not then it must have not been in the list of primes.
> go back to 2. and select the next smaller prime because the odd
> number that comes from (p-1)+1 isn't prime for that prime.
>
>
> Observation:
>
> This technique could be used to extrapolate potential unknown primes from
> known primes since it produces a much smaller list of potential candidates
> than simply testing consecutive odds via a sieve. It also is not as porous
> as Mersenne Prime tests.
Bullshit.
Your "technique" ASSUMES that you already know the "largest prime that
is smaller by at least 4".
> 2+(largest_known_prime-1)+(next_smallest_unknown_prime-1)=big_even_number
>
> So, we need a way of guestimating the magnitude of the next prime and pick
> big_even_numbers that are appropriate.
>
> Observation: For a given x the number of primes <x is limited by x/ln(x).
> So we could note the points where x/ln(x) increases by 1.
>
Your observation is incorrect.
Consider x = 8. x/ln(x) =~ 3.84, but the number of primes < 8 (2,3,5,7) is
4, that is more than 8/ln(8).
I suggest checking observations more carefully.
igor
> Even Number Sum's:
>
> 6, 2+2+2
> 8, 2+2+4
> 10, 2+2+6
> 12, 2+4+6
> 14, 2+6+6
> 18, 2+4+12
> 20, 2+2+16
> 22, 2+2+18
> 24, 2+4+18
> 26, 2+6+18
> 28, 2+4+22
> 30, 2+6+22
> 32, 2+2+28
> 34, 2+2+30
> 36, 2+4+30
> 38, 2+6+30
> 40, 2+2+36
>
> ...
>
> 100, 2+2+96
>
> ...
>
> 398, 2+8+388 Note: this breaks since 8 isn't available. 9 ain't a
> prime.
>
> 2+14+382 next smaller doesn't work since 14 isn't there.
> 15 isn't prime.
>
> 2+18+378 that one works!
>
> ...
>
> 666, 2+4+660
>
> ...
>
> 758, 2+6+750
>
> ...
>
> 1,032, 2+10+1020
>
> ...
>
> 1,044, 2+4+1038
>
> ...
>
> 6,236, 2+6+6228
>
> ...
>
> 7,920, 2+12+7906 Note: can't use 7,919 since it's delta is <4.
>
>
> Iterated Sums:
>
> 2+2+2=6
> 2+2+4=8
> 2+2+6=10
> 2+2+10=14
>
> 2+4+2=8
> 2+4+4=10
> 2+4+6=12
> 2+4+10=16
>
> 2+6+2=10
> 2+6+4=12
> 2+6+6=14
> 2+6+10=18
>
>
> 1st 1,000 Primes & their p-1's:
>
> p p-1
> -------------------------------
>
> 2 1
> 3 2
> 5 4
> 7 6
> 11 10
> 13 12
> 17 16
> 19 18
> 23 22
> 29 28
> 31 30
> 37 36
> 41 40
> 43 42
> 47 46
> 53 52
> 59 58
> 61 60
> 67 66
> 71 70
> 73 72
> 79 78
> 83 82
> 89 88
> 97 96
> 101 100
> 103
> 107
> 109
> 113
>
>
> 127 131 137 139 149 151 157 163 167 173
>
>
> 179 181 191 193 197 199 211 223 227 229
>
>
> 233 239 241 251 257 263 269 271 277 281
>
>
> 283 293 307 311 313 317 331 337 347 349
>
>
> 353 359 367 373 379 383 389 397 401 409
>
>
> 419 421 431 433 439 443 449 457 461 463
>
> 467 479 487 491 499 503 509 521 523 541
>
> 547 557 563 569 571 577 587 593 599 601
>
> 607 613 617 619 631 641 643 647 653 659
>
> 661 673 677 683 691 701 709 719 727 733
>
> 739 743 751 757 761 769 773 787 797 809
>
> 811 821 823 827 829 839 853 857 859 863
>
> 877 881 883 887 907 911 919 929 937 941
>
> 947 953 967 971 977 983 991 997 1009 1013
>
> 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069
>
> 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
>
> 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223
>
> 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291
>
> 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373
>
> 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451
>
> 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511
>
> 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583
>
> 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
>
> 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733
>
> 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811
>
> 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889
>
> 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987
>
> 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053
>
> 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129
>
> 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
>
> 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287
>
> 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357
>
> 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423
>
> 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531
>
> 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617
>
> 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687
>
> 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
>
> 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819
>
> 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903
>
> 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999
>
> 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079
>
> 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181
>
> 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257
>
> 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
>
> 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413
>
> 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511
>
> 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571
>
> 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643
>
> 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727
>
> 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821
>
> 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
>
> 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989
>
> 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057
>
> 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139
>
> 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231
>
> 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297
>
> 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409
>
> 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
>
> 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583
>
> 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657
>
> 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751
>
> 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831
>
> 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937
>
> 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003
>
> 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087
>
> 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179
>
> 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279
>
> 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387
>
> 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443
>
> 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521
>
> 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639
>
> 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693
>
> 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791
>
> 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857
>
> 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939
>
> 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053
>
> 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133
>
> 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221
>
> 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301
>
> 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367
>
> 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473
>
> 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571
>
> 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673
>
> 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761
>
> 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833
>
> 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917
>
> 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997
>
> 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103
>
> 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207
>
> 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297
>
> 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411
>
> 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499
>
> 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561
>
> 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643
>
> 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723
>
> 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829
>
> 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919
>
- Igor.