20071010, 14:08  #1 
Jun 2003
7×167 Posts 
Fischbach Representations.
These are my definitions based of Carl Fischbach's post.
A Fischbach representation of the first kind expresses a prime number p as a sum A + B or a difference A  B, where A and B (necessarily coprime) are smooth to some prime S, sqr(p) < S < p, and where all primes <= S are factors of one of A and B. Examples: 2+3=5 2*2+3=7 2+3*3=11 2*5+3=13 3*5+2=17 2*2*2*35=19 2*3*57=23 7*52*3=29 5*3*32*7=31 5*3*2+7=37 5*7+2*3=41 2*5*73*3*3=43 All of these were given by Fischbach himself with the exception of 11. In each case the corresponding S is the largest prime factor on the LHS. A Fischbach representation of the second kind permits A and/or B to be nonsmooth provided that any additional prime factors have Fischbach representations of the second kind, and there are no loops in their derivation. All firstkind representations are trivially also second kind representations. 3*5*72*29=47 In this case S=7 while 29 has been represented above without the use of 47. Further secondkind representations may now use 47. The problem is to continue Fischbach's list, using firstkind representations were possible. Are any primes not representable? 
20071010, 14:21  #2  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×3×311 Posts 
Quote:
Selfridge, et. al. It is very closely related to what you are trying to discuss. 

20071010, 14:48  #3 
Jan 2005
Transdniestr
503 Posts 
Extending the list
Couldn't you represent 47 as 5*7 + 2*2*3 because 7 > sqrt(47)?
47 = 7*11  2*3*5 53 = 3*5*11  2*2*2*2*7 61 = 3*5*7 2*2*11 83 = 3*5*7 2*11 97 = 2*3*7 + 5*11 Last fiddled with by grandpascorpion on 20071010 at 15:27 
20071010, 15:23  #4 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Let T be the smallest prime > sqrt(p), and d = 2*3*5*...*T, then type 1 representations a±b=p need
d  a*(p±b) They also need that a*(p±b) contains consecutive primes in its factorization. Code:
primorial(n) = {local(r, p); r=1;forprime(p=2,n,r*=p);return(r);} check(n, l) = {local(r, p); r=n; if(r==1, return(1)); forprime(p=2,l, if(r==1,return(1)); if(r%p!=0,return(0)); while(r%p==0,r/=p);)} forprime(p = 47, 500,T = nextprime(ceil(sqrt(p))); d = primorial(T); for (i = 1, p1, if((i*(pi)) % d == 0 && check(i*(pi),p), print(i, " + ", pi, " = ", p); break;)); for (i = 1, 1000000, if((i*(p+i)) % d == 0 && check(i*(p+i),p), print(p+i, "  ", i, " = ", p); break;));) 5 + 42 = 47 75  28 = 47 165  112 = 53 224  165 = 59 105  44 = 61 165  98 = 67 126  55 = 71 150  77 = 73 154  75 = 79 105  22 = 83 110  21 = 89 42 + 55 = 97 132  35 = 97 35 + 66 = 101 231  130 = 101 33 + 70 = 103 180  77 = 103 30 + 77 = 107 140  33 = 107 154  45 = 109 168  55 = 113 715  588 = 127 495  364 = 131 462  325 = 137 594  455 = 139 429  280 = 149 385  234 = 151 1092  935 = 157 273  110 = 163 440  273 = 167 13923  13750 = 173 4004  3825 = 179 1105  924 = 181 3740  3549 = 191 6160  5967 = 193 5202  5005 = 197 4840  4641 = 199 7735  7524 = 211 146523  146300 = 223 1547  1320 = 227 6664  6435 = 229 4160  3927 = 233 4914  4675 = 239 2145  1904 = 241 1560  1309 = 251 2805  2548 = 257 858  595 = 263 10829  10560 = 269 1155  884 = 271 12597  12320 = 277 1386  1105 = 281 3003  2720 = 283 17765  17472 = 293 29700  29393 = 307 8398  8085 = 313 31920  31603 = 317 8976  8645 = 331 326040  325703 = 337 16055  15708 = 347 336490  336141 = 349 29393  29040 = 353 134064  133705 = 359 119680  119301 = 379 200583  200200 = 383 97020  96577 = 443 30107  29640 = 467 207207  206720 = 487 120666  120175 = 491 33649  33150 = 499 A problem is that a and b are not bounded in the ab representations. The search could be sped up by solving a*(p±a) == 0 (mod d) and trying only those a. Alex Last fiddled with by akruppa on 20071010 at 16:20 Reason: removed erroneous p = nextprime(p + 1), updated list 
20071010, 15:42  #5  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
pointed out the unboundedness then. As an algorithm, the idea is useless. OTOH, the *existence* of such a representation is trivial. 

20071010, 16:16  #6 
"Nancy"
Aug 2002
Alexandria
4643_{8} Posts 
Bug: the last p = nextprime(p + 1) must be removed or the code skips every other prime. Sorry.
The smallest prime I didn't find a type 1 representation for is 311 now. Bob, I don't think these representations is particularly useful, but it was kinda fun to write some code to find them. I'm not convinced that a type 1 representation exists for all primes, though. When allowing type 2, they probably do. I haven't given it much thought yet. Maybe I can find the old thread. Alex 
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