1,2

Diophant's formula 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 input 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.

Eric W. Weisstein, Diophantine Equation, 2nd powers, MathWorld.

Eric W. Weisstein, Fibonacci identity, MathWorld.

T(n,1) = prime(4*n-2)*prime(4*n-1)-prime(4*n-3)*p(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)*p(4*n). T(n,4) = prime(4*n-3)*prime(4*n-1)-prime(4*n-2)*prime(4*n).

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.

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

Cf. A000040.

Sequence in context: A066434 A040932 A008685 this_sequence A141529 A022987 A023473

Adjacent sequences: A162153 A162154 A162155 this_sequence A162157 A162158 A162159

sign,tabf

Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jun 26 2009

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 16 2009