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Old 2021-11-18, 12:32   #23
robert44444uk
 
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Quote:
Originally Posted by Dr Sardonicus View Post

I haven't even begun to think about what governs the size of c1, but although I've seen a number of cases where c1 is fairly close to c2, I haven't noticed any instances where c1 > c2.
c2-c1 negative - these are common at low N. There are 3158 examples in N< 135536 but none after this, checked up to N=304000. Between N=13336 and N=304000 the least difference is c2-c1=6 demonstrated by 182978.

For N = 10 and N=34 , c1=0
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Old 2021-11-18, 17:21   #24
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Coming back to Goldbach.

It is proved that there is a prime between N/2 and N (Bertrand- Chebyshev), with N even

Using this prime residue technique, is it correct to paraphrase Goldbach to say it is conjected that there exists at least 1 prime p(i) between N/2 to N where N mod p(i) == a prime, p(ii) with p(ii) =< p(i) ? I think this is equivalent to stating that there are 2 primes which add up to N.

It should be possible to list N in terms of the prime mod primes discovered between N/2 and N (c2) to find out the factors of N where the count of c2 is very low compared to the size of N. I have done this to N=200000, and the lowest are as follows:

My interpretation of the data is that the lowest number of prime pairs that add up to N are predominately found when N factors as 2^x.prime, x from upwards. In this range, there are no small factors other than 2. In this example, the smallest factor is 263.

Code:
N	p >n/2	c2	ratio	factors
198728	8347	1008	0.12076195	2.2.2.24841
182012	7692	929	0.120774831	2.2.45503
144128	6217	751	0.120797812	2.2.2.2.2.2.2.2.563
181904	7685	929	0.120884841	2.2.2.2.11369
199328	8366	1013	0.121085345	2.2.2.2.2.6229
196754	8272	1004	0.121373308	2.98377
191834	8073	980	0.121392295	2.95917
195368	8215	998	0.121485088	2.2.2.2.4421
158428	6791	826	0.121631571	2.2.39607
198916	8353	1016	0.121632946	2.2.223.223
167438	7127	867	0.121650063	2.83719
173948	7381	898	0.121663731	2.2.43487
191024	8039	979	0.121781316	2.2.2.2.11939
188882	7964	970	0.121798091	2.94441
168068	7151	871	0.121801147	2.2.42017
179618	7610	927	0.121813403	2.89809
189872	8004	976	0.12193903	2.2.2.2.11867
196628	8266	1008	0.121945318	2.2.49157
176678	7493	914	0.121980515	2.88339
183916	7771	948	0.121992022	2.2.45979
174362	7400	903	0.122027027	2.87181
176024	7448	909	0.122046187	2.2.2.22003
199846	8387	1024	0.122093716	2.99923
189578	7993	976	0.122106843	2.94789
188834	7963	973	0.122190129	2.263.359
At the other end of the spectrum, smooth N predominate

Code:
N	p >n/2	c2	 ratio 	factors
2310	152	114	 0.7500000000 	2.3.5.7.11
660	54	41	 0.7592592593 	2.2.3.5.11
270	25	19	 0.7600000000 	2.3.3.3.5
78	9	7	 0.7777777778 	2.3.13
96	9	7	 0.7777777778 	2.2.2.2.2.3
300	27	21	 0.7777777778 	2.2.3.5.5
840	65	51	 0.7846153846 	2.2.2.3.5.7
144	14	11	 0.7857142857 	2.2.2.2.3.3
34	5	4	 0.8000000000 	2.17
42	5	4	 0.8000000000 	2.3.7
84	10	8	 0.8000000000 	2.2.3.7
168	16	13	 0.8125000000 	2.2.2.3.7
240	22	18	 0.8181818182 	2.2.2.2.3.5
390	33	27	 0.8181818182 	2.3.5.13
180	17	14	 0.8235294118 	2.2.3.3.5
48	6	5	 0.8333333333 	2.2.2.2.3
126	12	10	 0.8333333333 	2.3.3.7
630	49	41	 0.8367346939 	2.3.3.5.7
60	7	6	 0.8571428571 	2.2.3.5
66	7	6	 0.8571428571 	2.3.11
150	14	12	 0.8571428571 	2.3.5.5
330	28	24	 0.8571428571 	2.3.5.11
420	35	30	 0.8571428571 	2.2.3.5.7
90	10	9	 0.9000000000 	2.3.3.5
120	13	12	 0.9230769231 	2.2.2.3.5
10	2	2	 1.0000000000 	2.5
16	2	2	 1.0000000000 	2.2.2.2
36	4	4	 1.0000000000 	2.2.3.3
210	19	19	 1.0000000000 	2.3.5.7

Last fiddled with by robert44444uk on 2021-11-18 at 17:26
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Old 2021-11-18, 20:25   #25
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The Prime Number Theorem (PNT) gives the asymptotic estimate N/log(n) - (N/2)/log(N/2) which is approximately (1/2)*N/log(N)) for the number of primes between N/2 and N.

The Hardy-Littlewood conjectural asymptotic formula for the number of representations of N as the sum of two odd primes is 2*C2*N/log2(N)*\prod_{p\mid N}^{p\;odd}\frac{p-1}{p-2}

where the product is over the odd prime factors of N and C2 = 0.66016... is the "twin primes constant."

This is of smaller order than the number of primes in (N/2, N) by a factor of log(N).

Your observation about small prime factors is perfectly in line with the last factor in the Hardy-Littlewood formula, a product over the odd prime factors of N. That factor is biggest when N has a lot of small prime factors, and is smallest if N is a power of 2 (when the factor is the empty product, or 1). For N = 2x*p, p an odd prime, the factor is (p-1)/(p-2).

I'm still making no further progress with the residues N%p for p < N/2. As we have already seen, you can carve out intervals N/(k+1) < p < N/k for which N%p is guaranteed to be composite, but that's not nearly enough to get the count of prime residues down to something of order N/log2(N).


EDIT: I did come up with a handwaving argument that for large even N the number of prime residues N%p, 2 < p < N/2, is of order N/log2(N).

By the Prime Number Theorem, the number of primes \le x is asymptotic to x/log(x). So if p is a "large" prime, about 1/log(p) of the numbers < p are prime.

Now for sqrt(N) < p < N/2, log(p) only varies between (1/2)*log(N) and log(N) - log(2).

So, assuming, by wave of hand, that for a given p < N/2 not dividing N the residues 1 to p-1 N%p are all "equally likely," we guess that the number of prime residues for sqrt(N) < p < N/2 is of order

1/log(N)*N/log(N) or N/log2(N).

There are obviously no more than sqrt(N) primes p < sqrt(N), and sqrt(N) is of lower order than N/log2(N).

Last fiddled with by Dr Sardonicus on 2021-11-19 at 20:03 Reason: as indicated
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Old 2021-11-20, 13:55   #26
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Quote:
Originally Posted by robert44444uk View Post
The straight line solution at y=1, suggests x = 69.24 which corresponds to N= e^69.24 or about 1.18e30, still out of reach from a calculation perspective, but significant lower than the previous estimate.
Using 2^30 as the major set of prime factors in N suggests N= e^75.33 corresponding to 5e32. At this type of value of N, the ratio again drops below 1, last seen with N=20

The data below was obtained by running my program for about a week.

Quote:

N-1 ------------------- ratio
2147483647 1.15794911583994
559419490303 1.143415432
636728901631 1.142847684
817117528063 1.141814031
1377610760191 1.140112789
1403380563967 1.140059613
1918776639487 1.139185906
2073395462143 1.138948385
2266668990463 1.138736027
2369748205567 1.138647817
2814277320703 1.138180496
2910914084863 1.138120794
3110630064127 1.137940743
3310346043391 1.137813568
3619583688703 1.137604767


Last fiddled with by robert44444uk on 2021-11-20 at 14:00
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Old 2021-11-20, 14:55   #27
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Quote:
Originally Posted by robert44444uk View Post
Using 2^30 as the major set of prime factors in N suggests N= e^75.33 corresponding to 5e32. At this type of value of N, the ratio again drops below 1, last seen with N=20

The data below was obtained by running my program for about a week.
Quote:
N-1 ------------------- ratio
2147483647 1.15794911583994
559419490303 1.143415432
Is this your original "density ratio?"
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Old 2021-11-21, 11:00   #28
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Quote:
Originally Posted by Dr Sardonicus View Post
Is this your original "density ratio?"
Yes, the last of these is the lowest ratio, N-1 prime, seen once small N are taken out of the consideration.

Last fiddled with by robert44444uk on 2021-11-21 at 11:16
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