OFFSET
1,2
COMMENTS
The map takes 4 integers c, b, m and r and maps them onto four integers b*m-c*r, c*m+b*r, b*m+c*r and c*m-b*r, linked via (c^2+b^2)*(m^2+r^2) = (b*m-c*r)^2+(c*m+b*r)^2 = (b*m+c*r)^2+(c*m-b*r)^2.
Here, the inputs are four consecutive primes c=prime(4n-3), b=prime(4n-2), m=prime(4n-1) and r=prime(4n), and the four quadratic combinations which are the bases of the squares are placed into the n-th row of the table.
LINKS
Eric W. Weisstein, Diophantine Equation, 2nd powers, MathWorld.
Eric W. Weisstein, Fibonacci identity, MathWorld.
FORMULA
T(n,1) = prime(4*n-2)*prime(4*n-1) - prime(4*n-3)*prime(4*n).
T(n,2) = prime(4*n-3)*prime(4*n-1) + prime(4*n-2)*prime(4*n).
T(n,3) = prime(4*n-2)*prime(4*n-1) + prime(4*n-3)*prime(4*n).
T(n,4) = prime(4*n-3)*prime(4*n-1) - prime(4*n-2)*prime(4*n).
EXAMPLE
For n=3, the primes 23, 29, 31 and 37 are mixed via (23^2 + 29^2)*(31^2 + 37^2) = 48^2 + 1786^2 = 1750^2 + 360^2, and 48, 1786, 1750 and -360 from the right hand sides fill the third row of the table.
MAPLE
A162156 := proc(n, k) c := ithprime(4*n-3) ; b := nextprime(c) ; m := nextprime(b) ; r := nextprime(m) ; op(k, [b*m-c*r, c*m+b*r, b*m+c*r, c*m-b*r] ) ; end: seq(seq(A162156(n, k), k=1..4), n=1..20) ; # R. J. Mathar, Sep 16 2009
CROSSREFS
KEYWORD
sign,tabf,less
AUTHOR
Juri-Stepan Gerasimov, Jun 26 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Sep 16 2009
STATUS
approved