From: ichudov@Algebra.COM (Igor Chudov @ home)
Date: Wed, 25 Nov 1998 04:11:50 +0800
To: ravage@EINSTEIN.ssz.com (Jim Choate)
Subject: Re: A bit more on Goldbach's and Primes
In-Reply-To: <199811241801.MAA26092@einstein.ssz.com>
Message-ID: <199811241918.NAA00906@manifold.algebra.com>
MIME-Version: 1.0
Content-Type: text



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.