

A161681


Primes that are the difference between a cube and a square (conjectured values).


3



2, 7, 11, 13, 19, 23, 47, 53, 61, 67, 71, 79, 83, 89, 107, 109, 127, 139, 151, 167, 191, 193, 199, 223, 233, 239, 251, 271, 277, 293, 307, 359, 431, 433, 439, 463, 487, 499, 503, 547, 557, 587, 593, 599, 631, 647, 673, 683, 719, 727, 769, 797, 859, 887, 919
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OFFSET

1,1


COMMENTS

The primes found among the differences are sorted in ascending order and unique primes are then extracted. I call this a "conjectured" sequence since I cannot prove that somewhere on the road to infinity there will never exist an integer pair x,y such that x^3y^2 = 3,5,17,..., missing prime. For example, testing x^3y^2 for x,y up to 10000, the count of some duplicates are:
duplicate,count
7,2
11,2
47,3
431,7
503,7
1999,5
28279,11
Yet for 3,5,17,29,... I did not find even one.
[Comment from Charles R Greathouse IV, Nov 03 2009: 587 = 783^3  21910^2, 769 = 1025^3  32816^2, and 971 = 1295^3  46602^2 were skipped in the original.]
Conjecture: The number of primes in x^3y*2 is infinite.
Conjecture: The number of duplicates for a given prime is finite. Then there is the other side  the primes that are not in the sequence 3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, ... Looks like a lot of twin components here. Do these have an analytical form? Is there such a thing as a undecidable sequence?


LINKS



FORMULA

Integers x,y such that x^3y^2 = p where p is prime. The generation bound is 10000.


EXAMPLE

3^3  4^2 = 15^3  58^2 = 11.


PROG

(PARI) diffcubesq(n) =
{
local(a, c=0, c2=0, j, k, y);
a=vector(floor(n^2/log(n^2)));
for(j=1, n,
for(k=1, n,
y=j^3k^2;
if(ispseudoprime(y),
c++;
a[c]=y;
)
)
);
a=vecsort(a);
for(j=2, c/2,
if(a[j]!=a[j1],
c2++;
print1(a[j]", ");
if(c2>100, break);
)
);
}


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



