Primenumbers Yahoo Groups Here's some Goldbach separation data =============================================== Warren Smith Message 1 of 18 Sep 18, 2012 ----------------------------------------------- GOLDBACH SEPARATION FUNCTION The underlying "Goldbach separation" data in this table was computed by Tony D. Noe in https://oeis.org/A155765/b155765.txt https://oeis.org/A155764/b155764.txt https://oeis.org/A047160 https://oeis.org/wiki/User:Jason_Kimberley/A047160 Thanks to Maximilian Hasler for pointing it out. For N=1..61, the Nth record-high value of m is printed, where m is the least number>=0 such that both a+m and a-m are primes, and we keep incrementing a starting from 2 (until a new record-high m is found). The final column is f(a) = 0.769422*ln(a)^3. This is my empirical upper bound on m, which happens to be valid within the range of the table. It can be too high by as much as a factor of 4.715. [When m=0 it is infinitely too high, but I ignore that.] N a m f(a) 1, 2, 0, 0.2562370500 2, 4, 1, 2.049893073 3, 8, 3, 6.918386629 4, 22, 9, 22.72372535 5, 46, 15, 43.18159525 6, 121, 18, 84.86824603 7, 128, 21, 87.88911839 8, 136, 27, 91.22489509 9, 146, 33, 95.23484236 10, 238, 39, 126.0861302 11, 265, 42, 133.6608183 12, 286, 45, 139.2165668 13, 341, 48, 152.6127767 14, 344, 75, 153.3014591 15, 526, 87, 189.2302088 16, 904, 93, 242.6605587 17, 1114, 117, 265.6928937 18, 1691, 120, 315.9877223 19, 1736, 135, 319.3490428 20, 1751, 138, 320.4553082 21, 1781, 168, 322.6471859 22, 2476, 183, 367.1566981 23, 3097, 210, 399.6126911 24, 3551, 228, 420.3639088 25, 5131, 300, 479.7382423 26, 8504, 333, 569.9862949 27, 10342, 369, 607.7710100 28, 18526, 393, 730.1603524 29, 22564, 453, 775.0008925 30, 24776, 525, 796.8949511 31, 40072, 621, 915.9875978 32, 68707, 720, 1063.016969 33, 125903, 810, 1246.038365 34, 174913, 846, 1353.652005 35, 181267, 1086, 1365.690833 36, 371428, 1281, 1623.109918 37, 827576, 1305, 1946.678604 38, 936118, 1515, 1999.977075 39, 1054141, 1590, 2052.240939 40, 1159864, 1617, 2094.964675 41, 1353559, 1722, 2165.244879 42, 2591107, 1794, 2477.958278 43, 3759164, 1833, 2670.035444 44, 5454062, 1851, 2871.819721 45, 5969581, 2010, 2922.274876 46, 6363983, 2064, 2958.371376 47, 6639736, 2085, 2982.466788 48, 6791669, 2112, 2995.371977 49, 7183186, 2217, 3027.501431 50, 9122719, 2352, 3167.106409 51, 10786495, 2754, 3267.468091 52, 23848807, 2784, 3771.667809 53, 27072352, 3339, 3856.745467 54, 46330868, 3429, 4231.517031 55, 57989356, 4395, 4395.000000 56, 183234439, 4488, 5299.392017 57, 297456022, 5391, 5714.632343 58, 1010526289, 5430, 6857.998184 59, 1085549026, 5877, 6929.306766 60, 1387050394, 6195, 7177.091207 61, 2117398594, 7107, 7618.514999 =============================================== djbroadhurst Message 2 of 18 Sep 19, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, Warren Smith wrote: > The final column is f(a) = 0.769422*ln(a)^3. This is my empirical > upper bound on m The cube of a log seems reasonable: one log for each member of the pair of primes, on average, and then we throw in an extra log for Tony Noe's extremal cases, à la Cramér. David =============================================== WarrenS Message 3 of 18 Sep 19, 2012 ----------------------------------------------- > > The final column is f(a) = 0.769422*ln(a)^3. This is my empirical > > upper bound on m > > The cube of a log seems reasonable: one log for each member > of the pair of primes, on average, and then we throw in an > extra log for Tony Noe's extremal cases, à la Cramér. > > David --that was actually exactly my thinking. However, later, I thought some more. First of all, this picture https://oeis.org/wiki/User:Jason_Kimberley/A047160 shows what would seem to be a lot of mysterious nonrandom patterns. Think they are real? So if you believe in them, then Cramer-type randomness-based thinking is out the window, or at least might need to be weakened, e.g. maybe only 1 in every logN guys is "random" so you need a 4th power of log not a 3rd power? Second, I therefore played around some more trying to find better empirical upper bounding functions... I tried f(a) = C * ln(a-1)^B and the best parameters I found were B=3.0 and C=0.76942 which is an empirical upper bound tight within range of table to within a factor of 4.70. So, my initial guess seems confirmed in that sense -- although it remains unhappy in the sense 4.70 is a disappointingly wide range. Third, it would be better if the data this all is based on, were extended. For that, you'd need a fast prime generating sieve and some other fast code for, e.g. symbolically multiplying polynomials of degree 32768 using FFT (the polynomials are "generating functions" and the multiplication finds "prime coincidences") but I would think going out to a=10^14 should be feasible if anybody wanted to try hard. The present data only goes out to a few times 10^9. Perhaps somebody already did this computation though. =============================================== WarrenS Message 4 of 18 Sep 19, 2012 ----------------------------------------------- > this picture > https://oeis.org/wiki/User:Jason_Kimberley/A047160 > shows what would seem to be a lot of mysterious nonrandom patterns. > Think they are real? --Actually, I think these "patterns" simply arise from the presence of occasional big prime gaps. Whenever you have one, the two primes at each end of the gap tend to get used a lot in min-discrepancy Goldbach pairs, resulting is a symmetrical pair of "45-degree dotted lined segments" in that picture. Aside from that effect, it looks patternless. Hence the usual Cramer-type statistical model ought to be decently valid, hence my log-cubed bound ought to be decent. By the way, although I pretty much believe in Cramer-type thinking, there might be subtle lower-order nonrandom effects so that when, e.g. Cramer predicts the max prime gap below N is log(N)^2, really it ought to be, say, C*log(N)^2 for a non-obvious value of C, or log(N)^2 * logloglog(N), or something. I don't recommend trusting this kind of thinking to the utmost degree. But these kinds of anti-Cramer effects, if any, would be very difficult to spot computationally. =============================================== marku606 Message 5 of 18 Sep 21, 2012 ----------------------------------------------- Here's some other Goldbach separation data obtained by looking at the separation with the lowest prime divisors. 2N = p1 + p2. We look at the factors of p2 - p1 such that the largest prime which divides p2 - p1 is minimal. For instance, 4 = 2 + 2 ; 2 - 2 = 0 8 = 3 + 5 ; 5 - 3 = 2 16 = 5 + 11 ; 11 - 5 = 2*3 48 = 19 + 29 ; 29 - 19 = 2*5 Here are the results for 2n up to 100,000: [2n, minimal largest prime that divides p2-p1] [4, 0] [8, 2] [16, 3] [48, 5] [180, 7] [420, 11] [1680, 13] [21840, 17] [27300, 19] [78540, 23] The minimal largest prime seems to be approximated by log(2n)^1.3 but I have not looked very far. Mark =============================================== djbroadhurst Message 6 of 18 Sep 22, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, "marku606" wrote: > 2N = p1 + p2. We look at the factors of p2 - p1 such that > the largest prime which divides p2 - p1 is minimal. > Here are the results for 2n up to 100,000: > [2n, minimal largest prime that divides p2-p1] > [4, 0] [8, 2] [16, 3] [48, 5] [180, 7] [420, 11] > [1680, 13] [21840, 17] [27300, 19] [78540, 23] Definition: For prime p, let n(p) be the smallest integer n > 1 for which there is no pair of primes [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. Examples: [p, n(p), factor(n(p))] [2, 8, Mat([2, 3])] [3, 24, [2, 3; 3, 1]] [5, 90, [2, 1; 3, 2; 5, 1]] [7, 210, [2, 1; 3, 1; 5, 1; 7, 1]] [11, 840, [2, 3; 3, 1; 5, 1; 7, 1]] [13, 10920, [2, 3; 3, 1; 5, 1; 7, 1; 13, 1]] [17, 13650, [2, 1; 3, 1; 5, 2; 7, 1; 13, 1]] [19, 39270, [2, 1; 3, 1; 5, 1; 7, 1; 11, 1; 17, 1]] [23, 1492260, [2, 2; 3, 1; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]] Puzzle: Find n(29). Conjecture: n(p) is p-smooth for all prime p. David =============================================== djbroadhurst Message 7 of 18 Sep 23, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > Definition: For prime p, let n(p) be the smallest > integer n > 1 for which there is no pair of primes > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > Puzzle: Find n(29). Puzzle 31: Prove that n(31) < 3*10^8. David =============================================== djbroadhurst Message 8 of 18 Sep 23, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > Definition: For prime p, let n(p) be the smallest > integer n > 1 for which there is no pair of primes > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > Puzzle: Find n(29). > > Puzzle 31: Prove that n(31) < 3*10^8. Puzzle 37: Prove that n(37) < 2*10^10. Comment: This may suggest a "hard" new OEIS sequence, when the answers are in. David =============================================== Maximilian Hasler Message 9 of 18 Sep 23, 2012 ----------------------------------------------- On Sun, Sep 23, 2012 at 11:31 AM, djbroadhurst wrote: > > > Definition: For prime p, let n(p) be the smallest > > integer n > 1 for which there is no pair of primes > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > Puzzle 31: Prove that n(31) < 3*10^8. It seems that n(31) is the smallest member of a set including n = 281291010 = 29#/23 (?) Maximilian =============================================== Maximilian Hasler Message 10 of 18 Sep 23, 2012 ----------------------------------------------- > On Sun, Sep 23, 2012 at 11:31 AM, djbroadhurst wrote: >> >> > Definition: For prime p, let n(p) be the smallest >> > integer n > 1 for which there is no pair of primes >> > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. >> >> Puzzle: Find n(29). Hint: don't search beyond 11741730 = 23# / 19. Maximilian =============================================== djbroadhurst Message 11 of 18 Sep 23, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, Maximilian Hasler wrote: > > Definition: For prime p, let n(p) be the smallest > > integer n > 1 for which there is no pair of primes > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > > Puzzle 31: Prove that n(31) < 3*10^8. > > It seems that n(31) is the smallest member of a set > including n = 281291010 = 29#/23 (?) Indeed. Moreover, I have shown, by exhaustion, that 281291010 is the smallest value of n(31) consistent with: > Conjecture: n(p) is p-smooth for all prime p. for which my personal mnemonic is the "Kupferbach conjecture". The "Kupferbach" and Goldbach conjectures seem to me to be logically independent: a resolution of either of them does not seem to yield an immediate resolution of the other. While "Kupferbach" is supported by far less evidence, yet it might be easier to prove than Goldbach? David =============================================== djbroadhurst Message 12 of 18 Sep 23, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, Maximilian Hasler wrote: > > Definition: For prime p, let n(p) be the smallest > > integer n > 1 for which there is no pair of primes > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > > Puzzle: Find n(29). > > Hint: don't search beyond 11741730 = 23# / 19. Congrats for the correct answer. I had already proven, unconditionally, that n(29) = 11741730, so that's one more for OEIS, if Maximilian would like to draft an entry. With thanks to Mark for the idea, David =============================================== Mark Message 13 of 18 Sep 24, 2012 ----------------------------------------------- My thanks to David for taking an idea to the next level, brilliant. I ran some laggardly code last night and only this morning did I arrive at the 11741730 figure. I remember (was it last year) doing some investigations on the counts of p smooth numbers, and it was David who again took it to the next level by arriving at formulae to compute exact counts, if I recall. The King of Smooth in my mind. :) Mark --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > > > > --- In primenumbers@yahoogroups.com, > Maximilian Hasler wrote: > > > > Definition: For prime p, let n(p) be the smallest > > > integer n > 1 for which there is no pair of primes > > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. > > > > > Puzzle: Find n(29). > > > > Hint: don't search beyond 11741730 = 23# / 19. > > Congrats for the correct answer. I had already proven, > unconditionally, that n(29) = 11741730, so that's one > more for OEIS, if Maximilian would like to draft an entry. > > With thanks to Mark for the idea, > > David > =============================================== djbroadhurst Message 14 of 18 Sep 24, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, "Mark" wrote: > I ran some laggardly code last night and only this > morning did I arrive at the 11741730 figure. An OEIS sequence 8, 24, 90, 210, 840, 10920, 13650, 39270, 1492260, 11741730 has been proposed by Maximilian and awaits approval. Thereafter we have only the upper bounds n(31) <= 281291010 = 19#*29 n(37) <= 10919808900 = 5#*17#*23*31 where > Definition: For prime p, let n(p) be the smallest > integer n > 1 for which there is no pair of primes > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. The upper limit for p = 31 is tight on the fairly reasonable assumption that n(31) is 31-smooth. The upper limit for p = 37 is tight on the rather stronger assumption that n(37) is both 37-smooth and also divisible by 17#. It is instructive to study the Goldbach partitions, n +/- m, in the case n = 281291010, where there is no partition for which m is 31-smooth. The near misses are these 37-smooth values of m: [19573, [23, 2; 37, 1]] [450179, [23, 3; 37, 1]] [817811, [23, 1; 31, 2; 37, 1]] [1165019, [23, 1; 37, 3]] [1570243, [31, 1; 37, 3]] [1874161, Mat([37, 4])] [13955549, [23, 3; 31, 1; 37, 1]] [16656623, [23, 3; 37, 2]] [36115589, [23, 1; 31, 1; 37, 3]] Since we require gcd(m,n) = 1 for the primality of n +/- m, the only primes less than 37 available to m are 23 and 31, neither of which were used for this amusing Goldbach pair: 19#*29+37^4 is prime 19#*29-37^4 is prime It took Pari-GP about 2020 GHz-seconds to prove that there is no 37-smooth value of m in the case n = 10919808900. The near misses are these 41-smooth values of m: [52307677, [29, 2; 37, 1; 41, 2]] [66737381, [29, 1; 37, 2; 41, 2]] [76840601, [37, 4; 41, 1]] [197696957, [19, 4; 37, 1; 41, 1]] [1680914269, [29, 3; 41, 3]] [4493598401, [19, 4; 29, 2; 41, 1]] which may also use the primes 19, 29 and 37, omitted from n. Note that 19 and 37 do not appear here: 5#*17#*23*31+(29*41)^3 is prime 5#*17#*23*31-(29*41)^3 is prime David =============================================== Robert Gerbicz Message 15 of 18 Sep 25, 2012 ----------------------------------------------- "Thereafter we have only the upper bounds n(31) <= 281291010 = 19#*29 n(37) <= 10919808900 = 5#*17#*23*31" Using exhaustive search there is no better solution, so n(31)=**281291010 and n(37)=10919808900. (and also confirmed smaller n() values). First sieve out n, where 2n=p1+p2 and p1<=p2<=p1+bound and p2-p1 is p-smooth (you can choose say bound=20000). For the survivors used backtracking to generate possible values for d=p2-p1 (where d is p-smooth) and done primality tests for p1 and p2. It took 3 miutes to find n(31) and 12 hours for n(37). [Non-text portions of this message have been removed] =============================================== WarrenS Message 16 of 18 Sep 25, 2012 ----------------------------------------------- > Definition: For prime p, let n(p) be the smallest > integer n > 1 for which there is no pair of primes > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth. --Equivalent easier-to-read definition: For prime q, let n(q) be the least integer n>1 for which every pair of primes [p1, p2] with 2*n=p1+p2 has p2-p1 having a prime factor exceeding q. > Conjecture: n(q) is q-smooth for all prime q. > for which my personal mnemonic is the "Kupferbach conjecture". > The "Kupferbach" and Goldbach conjectures seem to me to be > logically independent: a resolution of either of them does > not seem to yield an immediate resolution of the other. --Note: if Goldbach conjecture is false, then the counterexample n, such that 2n is not the sum of two primes, will be the value of n(q) for all large-enough q. Thus Kupferbach will be asymptotically true. So no, they are not logically independent, at least as stated. (You may have intended a somewhat different statement.) > While "Kupferbach" is supported by far less evidence, > yet it might be easier to prove than Goldbach? > David --it might be a new approach to Goldbach which might say something interesting. I am not exactly an expert on the Goldbach conjecture but this Kupferbach thing is rather a different angle than anything Goldbach I saw before. I had wanted that Goldbach separation data because I happen to actually have a use for the Goldbach conjecture (!) and in that use, this "separation" way of quantifying it, is what matters. =============================================== djbroadhurst Message 17 of 18 Sep 25, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, "WarrenS" wrote: > --Note: if Goldbach conjecture is false, then the counterexample n, > such that 2n is not the sum of two primes, will be the value of n(q) > for all large-enough q. Thus Kupferbach will be asymptotically > true. "Kupferbach" conjectures that n(p) is p-smooth for /all/ prime p. A failure of Goldbach at N would tell us /nothing/ about "Kupferbach" in the cases with n(p) < N. Hence I maintain my stated opinion: > The "Kupferbach" and Goldbach conjectures seem to me to be > logically independent: a resolution of either of them does > not seem to yield an immediate resolution of the other. David =============================================== djbroadhurst Message 18 of 18 Sep 25, 2012 ----------------------------------------------- --- In primenumbers@yahoogroups.com, Robert Gerbicz wrote: > Using exhaustive search there is no better solution, > so n(31)=281291010 and n(37)=10919808900. Thanks, Robert. Now Maximilian can update http://oeis.org/A217016 David