Jacobi-Madden equation

The Jacobi-Madden equation is a Diophantine equation of form

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$with integer ${\displaystyle a,b,c,d\in \mathbb {Z} }$

This equation was first investigated by physicist Lee W. Jacobi and mathematician Daniel J. Madden in 2008. They were able to show that this equation has an infinite number of nontrivial solutions (all variables are non-zero).

history

The French mathematician Pierre de Fermat formulated Fermat's Great Theorem in the 17th century , which states that the following equation is unsolvable for positive integers : ${\displaystyle a,b,c,n\in \mathbb {N} ,n>2}$

${\displaystyle a^{n}+b^{n}=c^{n}}$

This theorem was proven by Andrew Wiles in 1994 , but Fermat was already able to provide a proof for the case using the method of infinite descent . Thus it was already known in the 17th century that the equation ${\displaystyle n=4}$

${\displaystyle a^{4}+b^{4}=c^{4}}$

has no integer nontrivial solution (i.e. with ). ${\displaystyle a,b,c\not =0}$

In 1769 Leonhard Euler formulated the following conjecture (the so-called Euler's conjecture ):

There are no positive integer solutions to the equation for . ${\displaystyle a_{1},a_{2},\ldots ,a_{n}\in \mathbb {N} }$ ${\displaystyle a_{1}^{n}+a_{2}^{n}+\dotsb +a_{n-1}^{n}=a_{n}^{n}}$ ${\displaystyle n\geq 3}$

Broken down to this assumption means: ${\displaystyle n=4}$

${\displaystyle a^{4}+b^{4}+c^{4}=e^{4}}$ has no positive integer solutions ${\displaystyle a,b,c,e\in \mathbb {N} }$

However, Euler could not find a proof or a counterexample for this conjecture . It was not until 1966 that a counterexample was discovered ( ), which made it clear that Euler's conjecture was wrong. In 1988, the then 22-year-old mathematician Noam Elkies found the following counterexample for : ${\displaystyle n\geq 4}$ ${\displaystyle n=5}$ ${\displaystyle 27^{5}+84^{5}+110^{5}+133^{5}=144^{5}}$ ${\displaystyle n=4}$

${\displaystyle 2682440^{4}[phone]^{4}[phone]^{4}=20615673^{4}}$

Noam Elkies was able to show as early as 1987 that there must be an infinite number of integer solutions to the equation . In 1988, while searching for the smallest solution to this equation, mathematician Roger Frye of Thinking Machines Corporation found after 100 hours of computer running time with the Connection Machine that it was: ${\displaystyle n=4}$ ${\displaystyle a^{4}+b^{4}+c^{4}=e^{4}}$

${\displaystyle 95800^{4}+217519^{4}+414560^{4}=422481^{4}}$

Euler also suspected that it should be possible to find four 4th powers, the sum of which is a 4th power. In fact, this assumption was only proven in 1911 by R. Norrie with the following numerical example:

${\displaystyle 30^{4}+120^{4}+272^{4}+315^{4}=353^{4}}$

The general form of this equation is:

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=e^{4}}$ with integer ${\displaystyle a,b,c,d,e\in \mathbb {Z} }$

Some solutions to this equation have now been found. But it wasn't until 2008 that Lee W. Jacobi and Daniel J. Madden were able to show that this Diophantine equation has an infinite number of positive integer solutions. In 1964, the mathematician Simcha Brudno, a survivor of the Dachau concentration camp , discovered the first mathematically particularly beautiful solution to this equation:

${\displaystyle 955^{4}+1770^{4}+(-2634)^{4}+5400^{4}={\big (}955+1770+(-2634)+5400{\big )}^{4}\qquad (=5491^{4})}$

Independently of this, Wroblewski also found the following solution around 1964:

${\displaystyle 7590^{4}+(-31764)^{4}+27385^{4}+48150^{4}={\big (}7590+(-31764)+27385+48150{\big )}^{4}\qquad (=51361^{4})}$

Thus the first two solutions to a special case of the equation have been found. Because Jacobi and Madden were able to show that this equation has an infinite number of nontrivial solutions (all variables are not equal to zero), this special case is called the Jacobi-Madden equation , which has the following form: ${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=e^{4}}$

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$ with integer ${\displaystyle a,b,c,d\in \mathbb {Z} }$

Find an infinite number of solutions from two solutions to the Jacobi-Madden equation

In order to determine an infinite number of solutions to this equation from two individual solutions of the Jacobi-Madden equation, Jacobi and Madden developed the following method, which is briefly outlined here:

You start with the Jacobi-Madden equation

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$

and the following identity :

${\displaystyle a^{4}+b^{4}+(a+b)^{4}=2\cdot (a^{2}+ab+b^{2})^{2}}$

Proof of this identity:

The following transformations are necessary to prove this identity:

${\displaystyle {\begin{array}{rcl}a^{4}+b^{4}+(a+b)^{4}&=&a^{4}+b^{4}+(a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4})\\&=&2\cdot (a^{4}+2a^{3}b+3a^{2}b^{2}+2ab^{3}+b^{4})\\&=&2\cdot {\big (}(a^{2})^{2}+2\cdot a^{2}\cdot ab+2\cdot a^{2}\cdot b^{2}+2\cdot ab\cdot b^{2}+(ab)^{2}+(b^{2})^{2}{\big )}\\&=&2\cdot (a^{2}+ab+b^{2})^{2}\end{array}}}$

With which this identity could be proven. ${\displaystyle \Box }$

If one adds the expression to the Jacobi-Madden equation on both sides , one obtains: ${\displaystyle (a+b)^{4}+(c+d)^{4}}$

${\displaystyle {\big (}a^{4}+b^{4}+(a+b)^{4}{\big )}+{\big (}c^{4}+d^{4}+(c+d)^{4}{\big )}=(a+b)^{4}+(c+d)^{4}+(a+b+c+d)^{4}}$

Now you apply the above identity twice on the left and once on the right:

${\displaystyle 2\cdot (a^{2}+ab+b^{2})^{2}+2\cdot (c^{2}+cd+d^{2})^{2}=2\cdot {\big (}(a+b)^{2}+(a+b)(c+d)+(c+d)^{2}{\big )}^{2}}$

If you divide this equation by , you get ${\displaystyle 2}$

${\displaystyle (a^{2}+ab+b^{2})^{2}+(c^{2}+cd+d^{2})^{2}={\big (}(a+b)^{2}+(a+b)(c+d)+(c+d)^{2}{\big )}^{2}}$

But this is an equation of the form with , and , because one is only interested in integer solutions, it is a Pythagorean triple . Jacobi and Madden have now calculated two Pythagorean triples, namely and, using the two solutions of the Jacobi-Madden equation already known at the time, namely von Brudno and von Wroblewski, respectively . With each of these two solutions and with the elliptic curve method , they showed that one can construct an infinite number of other integer solutions of the Jacobi-Madden equation, all of which are not equal to zero. ${\displaystyle A^{2}+B^{2}=C^{2}}$ ${\displaystyle A=a^{2}+ab+b^{2}}$ ${\displaystyle B=c^{2}+cd+d^{2}}$ ${\displaystyle C=(a+b)^{2}+(a+b)(c+d)+(c+d)^{2}}$ ${\displaystyle (a,b,c,d)=(955,-2634,1770,5400)}$ ${\displaystyle (a,b,c,d)=(7590,-31764,27385,48150)}$ ${\displaystyle (A,B,C)=(5334511,41850900,42189511)}$ ${\displaystyle (A,B,C)=(825471036,4386948475,4463935411)}$

Further minimal solutions to the Jacobi-Madden equation

Since one can construct an infinite number of further solutions from every integer solution of the Jacobi-Madden equation, so-called minimal solutions are particularly interesting, which cannot be calculated from known solutions. In addition to the two integral minimum solutions of the Jacobi-Madden equation already mentioned:

${\displaystyle 955^{4}+1770^{4}+(-2634)^{4}+5400^{4}={\big (}955+1770+(-2634)+5400{\big )}^{4}\qquad (=5491^{4})}$

${\displaystyle 7590^{4}+(-31764)^{4}+27385^{4}+48150^{4}={\big (}7590+(-31764)+27385+48150{\big )}^{4}\qquad (=51361^{4})}$

the mathematician Seiji Tomita announced two further minimal solutions to the Jacobi-Madden equation in August 2015:

${\displaystyle 1229559^{4}+(-1022230)^{4}+1984340^{4}+(-107110)^{4}={\big (}1229559-1022230+1984340-107110{\big )}^{4}\qquad (=2084559^{4})}$

${\displaystyle 561760^{4}+1493309^{4}+3597130^{4}+(-1953890)^{4}={\big (}561760+1493309+3597130-1953890{\big )}^{4}\qquad (=3698309^{4})}$

Web links

• Eric S. Rowland: Elliptic curves and integral solutions to A ⁴ + B ⁴ + C ⁴ = D ⁴ . October 1964, pp. 1-7 , accessed December 16, 2004 . 

Individual evidence

1. ↑ ^{a b} Lee W. Jacobi, Daniel J. Madden: On a ⁴ + b ⁴ + c ⁴ + d ⁴ = (a + b + c + d) ⁴ . The American Mathematical Monthly 115 (3), 2008, pp. 220-236 , accessed October 1, 2019 . 
2. ^ Mathematicians find new solutions to an ancient puzzle. Phys.org, March 14, 2008, accessed October 1, 2019 . 
3. ↑ Noam D. Elkies : On A ⁴ + B ⁴ + C ⁴ = D ⁴ . Mathematics of Computation 51 (184), October 1988, pp. 825-835 , accessed October 1, 2019 . 
4. ↑ ^{a b c} Variations on Euler's Conjecture. Neue Zürcher Zeitung , May 14, 2008, accessed on October 1, 2019 . 
5. ^ ^{A b} Eric W. Weisstein : Diophantine Equation - 4th Powers. Wolfram MathWorld , accessed October 1, 2019 . 
6. ^ Jaroslaw Wroblewski: Database of solutions to the Euler's equation. Retrieved October 1, 2019 . 
7. ↑ Simcha Brudno: A further example of A ⁴ + B ⁴ + C ⁴ + D ⁴ = E ⁴ . Mathematical Proceedings of the Cambridge Philosophical Society 60 (4), October 1964, pp. 1027-1028 , accessed October 1, 2019 . 
8. ↑ ^{a b} Cai Tianxin: The Book Of Numbers. World Scientific , 2017, p. 297 , accessed October 3, 2019 . 
9. ↑ More elliptic curves for a ⁴ + b ⁴ + c ⁴ + d ⁴ = (a + b + c + d) ⁴ . math overflow , accessed October 3, 2019 . 
10. ↑ Seiji Tomita: New solutions of a ⁴ + b ⁴ + c ⁴ + d ⁴ = (a + b + c + d) ⁴ . 2015, accessed October 3, 2019 .