mersenneforum.org Primes of the form 30*n^2-1
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 2020-02-23, 16:20 #1 enzocreti   Mar 2018 2·271 Posts Primes of the form 30*n^2-1 Because I found that the prime 5879 is in some mysterious way involved in pg primes And realizing that 5879 is a prime of the form 30*n^2-1, I wonder if in oesi there is a sequence of primes of this form
 2020-02-23, 17:14 #2 carpetpool     "Sam" Nov 2016 5×67 Posts You could always make one yourself if it doesn't exist: Code: (09:13) gp > g=[]; for(i=1,300, if(isprime(30*i^2-1), g=concat(g,i))) (09:13) gp > g %5 = [1, 3, 4, 10, 14, 17, 18, 21, 22, 25, 29, 34, 35, 38, 42, 43, 48, 52, 55, 56, 60, 62, 63, 67, 73, 74, 78, 80, 90, 92, 94, 95, 99, 101, 105, 108, 113, 116, 118, 119, 126, 127, 129, 130, 132, 133, 139, 140, 143, 147, 153, 154, 157, 178, 181, 183, 186, 190, 192, 195, 199, 200, 207, 213, 216, 221, 224, 225, 232, 234, 238, 242, 244, 248, 251, 256, 259, 265, 269, 273, 276, 281, 284, 288, 290, 294, 297, 300] (09:13) gp >
 2020-02-24, 04:17 #3 CRGreathouse     Aug 2006 135448 Posts For a more general approach: Code: polyToSeqPrimes(p)=my(v=List(),len,x=variable(p),n,t); while(len<=260, if(isprime(t=subst(p,x,n)), listput(v,t); len+=#Str(t)+2); n++); v=Vec(v); v[1..#v-1]; polyToSeqIndices(p)=my(v=List(),len,x=variable(p),n); while(len<=260, if(isprime(subst(p,x,n)), listput(v,n); len+=#Str(n)+2); n++); v=Vec(v); v[1..#v-1]; polyToPrimeBFile(p)=my(v=vector(10^4),x=variable(p),n,i,t); while(i<#v, if(isprime(t=subst(p,x,n)), v[i++]=t); n++); v; polyToIndexBFile(p)=my(v=vector(10^4),x=variable(p),n,i); while(i<#v, if(isprime(subst(p,x,n)), v[i++]=n); n++); v; polyToSeqPrimes(30*'n^2-1) polyToSeqIndices(30*'n^2-1) Just enter the polynomial desired and get the primes and their indices computed for you. (Versions for 10,000 term b-files included as well.)
2020-02-25, 10:47   #4
enzocreti

Mar 2018

21E16 Posts
...

Quote:
 Originally Posted by carpetpool You could always make one yourself if it doesn't exist: Code: (09:13) gp > g=[]; for(i=1,300, if(isprime(30*i^2-1), g=concat(g,i))) (09:13) gp > g %5 = [1, 3, 4, 10, 14, 17, 18, 21, 22, 25, 29, 34, 35, 38, 42, 43, 48, 52, 55, 56, 60, 62, 63, 67, 73, 74, 78, 80, 90, 92, 94, 95, 99, 101, 105, 108, 113, 116, 118, 119, 126, 127, 129, 130, 132, 133, 139, 140, 143, 147, 153, 154, 157, 178, 181, 183, 186, 190, 192, 195, 199, 200, 207, 213, 216, 221, 224, 225, 232, 234, 238, 242, 244, 248, 251, 256, 259, 265, 269, 273, 276, 281, 284, 288, 290, 294, 297, 300] (09:13) gp >

I note that there are pairs (21,22) (42,43) (126,127) with 21,42,126 multiple of 7

Is it unlikely that these consecutive (a,b) are infinitely many?

2020-02-25, 10:51   #5
enzocreti

Mar 2018

2×271 Posts
...

Quote:
 Originally Posted by carpetpool You could always make one yourself if it doesn't exist: Code: (09:13) gp > g=[]; for(i=1,300, if(isprime(30*i^2-1), g=concat(g,i))) (09:13) gp > g %5 = [1, 3, 4, 10, 14, 17, 18, 21, 22, 25, 29, 34, 35, 38, 42, 43, 48, 52, 55, 56, 60, 62, 63, 67, 73, 74, 78, 80, 90, 92, 94, 95, 99, 101, 105, 108, 113, 116, 118, 119, 126, 127, 129, 130, 132, 133, 139, 140, 143, 147, 153, 154, 157, 178, 181, 183, 186, 190, 192, 195, 199, 200, 207, 213, 216, 221, 224, 225, 232, 234, 238, 242, 244, 248, 251, 256, 259, 265, 269, 273, 276, 281, 284, 288, 290, 294, 297, 300] (09:13) gp >

I note that there are pairs (21,22) (42,43) (126,127) (62,63)...(a,b) with either a or b multiple of 7...do you believe these pairs with consecutive a,b are infinite?

I call a(n) the n-th term of the sequence

so it seems that there are infinitely many terms a(n) such that a(n+1)=a(n)+1

a(n)=21 a(n+1)=23=a(n)+1

Last fiddled with by enzocreti on 2020-02-25 at 11:15

2020-03-01, 02:10   #6
carpetpool

"Sam"
Nov 2016

1010011112 Posts

Quote:
 Originally Posted by enzocreti I note that there are pairs (21,22) (42,43) (126,127) with 21,42,126 multiple of 7 Is it unlikely that these consecutive (a,b) are infinitely many?

We cannot prove or disprove this at the moment, but it should be implied there exist infinitely many pairs by one of Dickinson's conjecture. The pattern continues:

(18:08) gp > g=[]; for(i=1,2000, if(isprime(30*i^2-1) & isprime(30*i^2+60*i+29) & i%7==0, g=concat(g,[[i,i+1]])))
(18:09) gp > g
%12 = [[21, 22], [42, 43], [126, 127], [224, 225], [399, 400], [455, 456], [469, 470], [686, 687], [693, 694], [742, 743], [777, 778], [833, 834], [1057, 1058], [1092, 1093], [1127, 1128], [1309, 1310], [1484, 1485], [1659, 1660], [1673, 1674], [1750, 1751], [1792, 1793], [1806, 1807]]

 Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 sweety439 179 2023-08-08 19:54 enzocreti enzocreti 2 2020-02-20 08:46 enzocreti enzocreti 0 2020-02-17 16:28 a1call Miscellaneous Math 6 2018-12-11 03:34 carpetpool carpetpool 3 2017-01-26 01:29

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