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Amicable Quadruple


An amicable quadruple as a quadruple (a,b,c,d) such that

 sigma(a)=sigma(b)=sigma(c)=sigma(d)=a+b+c+d,
(1)

where sigma(n) is the divisor function.

If (a,b) and (x,y) are amicable pairs and

 GCD(a,x)=GCD(a,y)=GCD(b,x)=GCD(b,y)=1,
(2)

then (ax,ay,bx,by) is an amicable quadruple. This follows from the identity

 sigma(ax)=sigma(a)sigma(x)=(a+b)(x+y)=ax+ay+bx+by.
(3)

The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800).

Large amicable quadruples can be generated using the formula

 [a; b; c; d]=C_n[173·1933058921·149·103540742849; 173·1933058921·15531111427499; 336352252427·149·103540742849; 336352252427·15531111427499],
(4)

where

 C_n=2^(n-1)M_n·5^9·7^2·11^4·17^2·19·29^2·67·71^2·109·131·139·179·307·431·521·653·1019·1279·2557·3221·5113·5171·6949
(5)

and M_n is a Mersenne prime with n a prime >3 (Y. Kohmoto; Guy 1994, p. 59).


See also

Amicable Pair, Amicable Triple

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References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 59, 1994.

Referenced on Wolfram|Alpha

Amicable Quadruple

Cite this as:

Weisstein, Eric W. "Amicable Quadruple." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmicableQuadruple.html

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