Primes of the form 30*n^2-1

[enzocreti]

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

[carpetpool]

You could always make one yourself if it doesn't exist:

(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 >

[CRGreathouse]

For a more general approach:

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.)

[enzocreti]

Originally Posted by carpetpool

You could always make one yourself if it doesn't exist:

(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?

[enzocreti]

Originally Posted by carpetpool

You could always make one yourself if it doesn't exist:

(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

[carpetpool]

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]]