 mersenneforum.org Can you find another number like 2200?
 Register FAQ Search Today's Posts Mark Forums Read  2018-12-03, 02:37 #1 goldbug   Dec 2018 2·11 Posts Can you find another number like 2200? Here is something I am having trouble with related to Goldbach Conjecture and maybe someone has some ideas on how to improve the search? I think these numbers will be exceedingly rare if they exist at all. Can anyone find another even number and two primes like 2200,3, and 13? 2n=2200 p1=3 p2=13 2n-p1=2197=p2^3 2n-p2=2187=p1^7 2n minus each prime equals the other prime to a power. This is the only example I have found, but I haven't checked very far (100000). It gets combinatorically hard to search pretty quickly so I would rather search smarter. It is fairly easy to show there are no single prime patterns like this and I would like to extend the search to 3,4, etc primes as well where each of the differences composed only of powers of the other primes.   2018-12-03, 06:26 #2 Batalov   "Serge" Mar 2008 Phi(4,2^7658614+1)/2 59×157 Posts Ok, you are asking for a N = an + b = bm + a. Then an - a = bm - b. So: make a pile of an - a with various a and n, and wait for a collision.   2018-12-03, 06:26 #3 uau   Jan 2017 10101102 Posts If you reformulate a problem just a little bit, it does not actually become combinatorially hard. 3^7 + 13 = 3 + 13^3 (= 2200) 3^7 - 3 = 13^3 - 13 (= 2184) So you don't actually need to consider pairs of primes separately. Just calculate "p^n - p" for all primes p and powers n >= 2 such that the value is less than your limit, and see if you get the same number twice. To check this for all numbers up to some limit, you only need to consider primes up to about sqrt(limit) as otherwise prime^2 would already be too big. I checked that 2184 is the only such number up to about 10000000000000000 (a hundred million squared, considering primes up to hundred million). That took less than half a minute with a Python script.   2018-12-03, 13:14 #4 ATH Einyen   Dec 2003 Denmark 22·3·251 Posts If a,b has to be prime: 32 - 3 = 23 - 2 133 - 13 = 37 - 3 otherwise: 62 - 6 = 25 - 2 152 - 15 = 63 - 6 162 - 16 = 35 - 3 912 - 91 = 213 - 2 2802 - 280 = 57 - 5 49302 - 4930 = 305 - 30 Last fiddled with by ATH on 2018-12-03 at 13:14   2018-12-03, 16:35   #5
science_man_88

"Forget I exist"
Jul 2009
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20C016 Posts Quote:
 Originally Posted by goldbug 2n minus each prime equals the other prime to a power.

let p1 be p, let p2 be q there are 4 main cases at play (assuming p,q aren't 2):

q is 3 mod 4, is raised to an odd power, and n is odd, leading to p is 3 mod 4

q is 3 mod 4, is raised to an odd power, and n is even, leading to p is 1 mod 4

q is raised to an even power, and n is odd, leading to p is 1 mod 4

q is raised to an even power, and n is even, leading to p is 3 mod 4

now just put it into practice in code.   2018-12-03, 16:49 #6 Dr Sardonicus   Feb 2017 Nowhere 105316 Posts If gcd(a,b) = 1 then b divides a^(n-1) - 1, and a divides b^(m-1) - 1, so znorder(Mod(b,a)) divides n-1 and znorder(Mod(a,b)) divides m-1. This might tend to push up the possible values of m and n for a given a and b. (One way to keep at least one of the znorders small is if a divides b-1, assuming a < b. If m <> n the equation an - bm = a - b is curious, in that (1) if m and n are greater than 1, the difference in powers is quite small, and (2) with m and n different, the difference being divisible by a - b is curious.   2018-12-03, 18:32 #7 goldbug   Dec 2018 2210 Posts Thank you so much and the challenge continues Thanks batalov and uau for the collision suggestion and to uau for running up to 1M^2?!? This suggestion is going to help me greatly in extending the search to triplets, quadruplets of primes. Ideally it would be amazing to show that numbers of this form do not exist, or at least above some threshold. Can you see why? I will post here if I find any. Although, the triplets have a different form so the collision method becomes a little combinatoric as we generalize the problem upward. 2n-p1=p2^a*p3^b 2n-p2=p1^c*p3^d 2n-p3=p1^e*p2^f   2018-12-03, 18:46   #8
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts Quote:
 Originally Posted by goldbug Thanks batalov and uau for the collision suggestion and to uau for running up to 1M^2?!? This suggestion is going to help me greatly in extending the search to triplets, quadruplets of primes. Ideally it would be amazing to show that numbers of this form do not exist, or at least above some threshold. Can you see why? I will post here if I find any. Although, the triplets have a different form so the collision method becomes a little combinatoric as we generalize the problem upward. 2n-p1=p2^a*p3^b 2n-p2=p1^c*p3^d 2n-p3=p1^e*p2^f
my cases can be generalized to any number of prime powers...   2018-12-03, 19:00 #9 goldbug   Dec 2018 2·11 Posts Yes I think the cases you describe will be useful for cases of 3 or more primes since it will reduce the number of triplets that need to be checked. I still haven't wrapped my head around how to extend the collision method to triplets and beyond but it seems like the problem explodes again a bit anyway.   2018-12-03, 19:02   #10
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts Quote:
 Originally Posted by goldbug Yes I think the cases you describe will be useful for cases of 3 or more primes since it will reduce the number of triplets that need to be checked. I still haven't wrapped my head around how to extend the collision method to triplets and beyond but it seems like the problem explodes again a bit anyway.
my cases generalize to multiplicities of primes of form 4k+3 being odd or even in the product.   2018-12-04, 00:08 #11 goldbug   Dec 2018 2·11 Posts #GoldbugNumbers Let me know if this holds water. The numbers I am searching for satisfy the following property. Maybe there is an easier way to search besides looking each order k=2,3,4,... separately? Given an even number 2n there exists some subset of the prime non-divisors of n 2 Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post MattcAnderson Puzzles 10 2017-05-21 01:52 allasc Math 0 2010-12-27 13:37 nibble4bits Puzzles 18 2006-01-07 10:40 Unregistered Math 11 2004-11-30 22:53 Fusion_power Puzzles 8 2003-11-18 19:36

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