OFFSET
1,2
REFERENCES
Joong Fang, Abstract Algebra, Schaum, 1963, Page 76.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
n=14: 29│ │421
n+(n+1)=14+(14+1)=29 14---196
n^2+(n+1)^2=196+225=421 │ X │
n+(n+1)^2=14+225=239 15---225 *15+225+1=241
n^2+(n+1)=196+15=211 211/ \239
.
n=5: 11│ │61
n+(n+1)=5+(5+1)=11 5---25
n^2+(n+1)^2=25+36=61 │ X │
n+(n+1)^2=5+36=41 6---36 *6+36+1=43
n^2+(n+1)=25+6=31 31/ \41
.
n=495: 991│ │491041
n+(n+1)=495+496=991 495---245025
n^2+(n+1)^2=491041 │ X │
n+(n+1)^2=246511 496---246016
n^2+(n+1)=245521 245521/ \246511
.
They form the group:
o 1 2 3 (i)
1 0 3 2
2 3 1 0
3 2 0 1
.
For example, for n=99:
99 9801 0 1 2 3 (i)
100 10000
9801 99 1 0 3 2
10000 100
10000 100
99 9801 2 3 1 0
100 10000 3 2 0 1
9801 99
The sum of each column and the sum of each diagonal is a prime number.
MATHEMATICA
Select[Range[51000], AllTrue[{#+(#+1), #^2+(#+1)^2, #+(#+1)^2, #^2+#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 21 2017 *)
PROG
(PARI)
is(n) = { isprime(n+(n+1)) & isprime(n^2+(n+1)^2) & isprime(n+(n+1)^2) & isprime(n^2+(n+1)); }
for(n=1, 10^6, if (is(n), print1(n, ", ")));
/* Joerg Arndt, Mar 26 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
César Aguilera, Mar 15 2013
STATUS
approved