|
Search: id:A111252
|
|
|
| A111252 |
|
Primes p such that the difference between the closest squares surrounding p is prime. |
|
+0
2
|
|
| 2, 3, 5, 7, 11, 13, 29, 31, 37, 41, 43, 47, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 331, 337, 347, 349, 353, 359, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 541, 547, 557, 563
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
Conjecture: The number of terms in this sequence is infinite.
That there are infinitely many terms in this sequence would follow from the Legendre conjecture (one of the Landau problems - see the Weisstein link) that there is always a prime between n^2 and (n+1)^2. This is still an open problem. - Max Alekseyev (maxale(AT)gmail.com), Apr 20 2006
|
|
LINKS
|
Eric Weisstein's World of Mathematics, Landau's Problems
|
|
FORMULA
|
Let p be a prime number and r = floor(sqrt(p)). Then the closest surrounding squares of p are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1. If if d is prime then list p.
|
|
EXAMPLE
|
29 is a prime number. 5^2 and 6^2 are the closest squares surrounding 29. Now the difference, 36-25 = 11 is prime so 29 is in the table.
|
|
MATHEMATICA
|
Clear[f, lst, p, n]; f[n_]:=IntegerPart[Sqrt[n]]; lst={}; Do[p=Prime[n]; If[PrimeQ[a=(f[p]+1)^2-f[p]^2], AppendTo[lst, p]], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 05 2009]
|
|
PROGRAM
|
(PARI) surrsqpr(n) = { local(x, y, j, r, d); forprime(x=2, n, r=floor(sqrt(x)); d=r+r+1; if(isprime(d), print1(p", ") ) ) }
|
|
CROSSREFS
|
Sequence in context: A079149 A024694 A024320 this_sequence A082843 A162567 A092728
Adjacent sequences: A111249 A111250 A111251 this_sequence A111253 A111254 A111255
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Cino Hilliard (hillcino368(AT)gmail.com), Nov 12 2005
|
|
|
Search completed in 0.002 seconds
|
