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Bhargava's Theorem


Let the nth composition of a function f(x) be denoted f^((n))(x), such that f^((0))(x)=x and f^((1))(x)=f(x). Denote the composition of f and g by f degreesg(x)=f(g(x)), and define

 sumF(a,b,c)=F(a,b,c)+F(b,c,a)+F(c,a,b).
(1)

Let

u=(a,b,c)
(2)
|u|=a+b+c
(3)
||u||=a^4+b^4+c^4,
(4)

and

f(u)=(a(b-c),b(c-a),c(a-b))
(5)
g(u)=(suma^2b,sumab^2,3abc).
(6)

Then if |u|=0 (i.e., c=-a-b),

||f^((m)) degreesg^((n))(u)||=||g^((n)) degreesf^((m))(u)||
(7)
=2(ab+bc+ca)^(2^(m+1)3^n),
(8)

where m,n in {0,1,...} and composition is done in terms of components.


See also

Diophantine Equation--4th Powers, Ford's Theorem

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References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 97-100, 1994.Bhargava, S. "On a Family of Ramanujan's Formulas for Sums of Fourth Powers." Ganita 43, 63-67, 1992.

Referenced on Wolfram|Alpha

Bhargava's Theorem

Cite this as:

Weisstein, Eric W. "Bhargava's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BhargavasTheorem.html

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