Diophantine Equation--Quartic

Call an equation involving quartics $m$-$n$ if a sum of $m$ quartics is equal to a sum of $n$ fourth Powers. The 2-1 equation

$$A^4+B^4=C^4$$ (1)

is a case of Fermat's Last Theorem with $n=4$ and therefore has no solutions. In fact, the equations

$$A^4\pm B^4=C^2$$ (2)

also have no solutions in Integers.

Parametric solutions to the 2-2 equation

$$A^4+B^4=C^4+D^4$$ (3)

are known (Euler 1802; Gérardin 1917; Guy 1994, pp. 140-141). A few specific solutions are

$$59^4+ 158^4 = 133^4+ 134^4 = 635{,}318{,}657$$ (4)

$$7^4+ 239^4 = 157^4+ 227^4 = 3{,}262{,}811{,}042$$ (5)

$$193^4+ 292^4 = 256^4+ 257^4 = 8{,}657{,}437{,}697$$ (6)

$$298^4+ 497^4 = 271^4+ 502^4 = 68{,}899{,}596{,}497$$ (7)

$$514^4+ 359^4 = 103^4+ 542^4 = 86{,}409{,}838{,}577$$ (8)

$$222^4+ 631^4 = 503^4+ 558^4 = 160{,}961{,}094{,}577$$ (9)

$$21^4+ 717^4 = 471^4+ 681^4 = 264{,}287{,}694{,}402$$ (10)

$$76^4+1203^4 = 653^4+1176^4 = 2{,}094{,}447{,}251{,}857$$ (11)

$$997^4+1342^4 = 878^4+1381^4 = 4{,}231{,}525{,}221{,}377$$ (12)

(Sloane's A003824 and A018786; Richmond 1920, Leech 1957), the smallest of which is due to Euler. Lander et al. (1967) give a list of 25 primitive 2-2 solutions. General (but incomplete) solutions are given by

$$x = a+b$$ (13)

$$y = c-d$$ (14)

$$u = a-b$$ (15)

$$v = c+d,$$ (16)

where

$$a = n(m^2+n^2)(-m^4+18m^2n^2-n^4)$$ (17)

$$b = 2m(m^6+10m^4n^2+m^2n^4+4n^6)$$ (18)

$$c = 2n(4m^6+m^4n^2+10m^2n^4+n^6)$$ (19)

$$d = m(m^2+n^2)(-m^4+18m^2n^2-n^4)$$ (20)

(Hardy and Wright 1979).

In 1772, Euler proposed that the 3-1 equation

$$A^4+B^4+C^4=D^4$$ (21)

had no solutions in Integers (Lander et al. 1967). This assertion is known as the Euler Quartic Conjecture. Ward (1948) showed there were no solutions for $D\leq 10,000$, which was subsequently improved to $D\leq 220,000$ by Lander et al. (1967). However, the Euler Quartic Conjecture was disproved in 1987 by Noam D. Elkies, who, using a geometric construction, found

$$2{,}682{,}440^4+15{,}365{,}639^4+18{,}796{,}760^4=20{,}615{,}673^4$$ (22)

and showed that infinitely many solutions existed (Guy 1994, p. 140). In 1988, Roger Frye found

$$95{,}800^4+217{,}519^4+414{,}560^4=422{,}481^4$$ (23)

and proved that there are no solutions in smaller Integers (Guy 1994, p. 140). Another solution was found by Allan MacLeod in 1997,

$$638{,}523{,}249^4 = 630{,}662{,}624^4 + 275{,}156{,}240^4 + 219{,}076{,}465^4.eqno$$ (24)

It is not known if there is a parametric solution.

In contrast, there are many solutions to the 3-1 equation

$$A^4+B^4+C^4=2D^4$$ (25)

(see below).

Parametric solutions to the 3-2 equation

$$A^4+B^4=C^4+D^4+E^4$$ (26)

are known (Gérardin 1910, Ferrari 1913). The smallest 3-2 solution is

$$3^4+5^4+8^4=7^4+7^4$$ (27)

(Lander et al. 1967).

Ramanujan gave the 3-3 equations

$$2^4+4^4+ 7^4 = 3^4+6^4+ 6^4$$ (28)

$$3^4+7^4+ 8^4 = 1^4+2^4+ 9^4$$ (29)

$$6^4+9^4+12^4 = 2^4+2^4+13^4$$ (30)

(Berndt 1994, p. 101). Similar examples can be found in Martin (1896). Parametric solutions were given by Gérardin (1911).

Ramanujan also gave the general expression

$$3^4+(2x^4-1)^4+(4x^5+x)^4 = (4x^4+1)^4+(6x^4-3)^4+(4x^5-5x)^4$$ (31)

(Berndt 1994, p. 106). Dickson (1966, pp. 653-655) cites several Formulas giving solutions to the 3-3 equation, and Haldeman (1904) gives a general Formula.

The 4-1 equation

$$A^4+B^4+C^4+D^4=E^4$$ (32)

has solutions

$$30^4+ 120^4+ 272^4+ 315^4 = 353^4$$ (33)

$$240^4+ 340^4+ 430^4+ 599^4 = 651^4$$ (34)

$$435^4+ 710^4+1384^4+2420^4 = 2487^4$$ (35)

$$1130^4+1190^4+1432^4+2365^4 = 2501^4$$ (36)

$$850^4+1010^4+1546^4+2745^4 = 2829^4$$ (37)

$$2270^4+2345^4+2460^4+3152^4 = 3723^4$$ (38)

$$350^4+1652^4+3230^4+3395^4 = 3973^4$$ (39)

$$205^4+1060^4+2650^4+4094^4 = 4267^4$$ (40)

$$1394^4+1750^4+3545^4+3670^4 = 4333^4$$ (41)

$$699^4+ 700^4+2840^4+4250^4 = 4449^4$$ (42)

$$380^4+1660^4+1880^4+4907^4 = 4949^4$$ (43)

$$1000^4+1120^4+3233^4+5080^4 = 5281^4$$ (44)

$$410^4+1412^4+3910^4+5055^4 = 5463^4$$ (45)

$$955^4+1770^4+2634^4+5400^4 = 5491^4$$ (46)

$$30^4+1680^4+3043^4+5400^4 = 5543^4$$ (47)

$$1354^4+1810^4+4355^4+5150^4 = 5729^4$$ (48)

$$542^4+2770^4+4280^4+5695^4 = 6167^4$$ (49)

$$50^4+ 885^4+5000^4+5984^4 = 6609^4$$ (50)

$$1490^4+3468^4+4790^4+6185^4 = 6801^4$$ (51)

$$1390^4+2850^4+5365^4+6368^4 = 7101^4$$ (52)

$$160^4+1345^4+2790^4+7166^4 = 7209^4$$ (53)

$$800^4+3052^4+5440^4+6635^4 = 7339^4$$ (54)

$$2230^4+3196^4+5620^4+6995^4 = 7703^4$$ (55)

(Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964, Lander et al. 1967), but it is not known if there is a parametric solution (Guy 1994, p. 139).

Ramanujan gave the 4-2 equation

$$3^4+9^4=5^4+5^4+6^4+8^4,$$ (56)

and the 4-3 identities

$$2^4+2^4+7^4 = 4^4+4^4+5^4+6^4$$ (57)

$$3^4+9^4+14^4 = 7^4+8^4+10^4+13^4$$ (58)

$$7^4+10^4+13^4 = 5^4+5^4+6^4+14^4$$ (59)

(Berndt 1994, p. 101). Haldeman (1904) gives general Formulas for 4-2 and 4-3 equations.

There are an infinite number of solutions to the 5-1 equation

$$A^4+B^4+C^4+D^4+E^4=F^4.$$ (60)

Some of the smallest are

$$2^4+ 2^4+ 3^4+ 4^2+ 4^2 = 5^4$$ (61)

$$4^4+ 6^4+ 8^4+ 9^4+14^4 = 15^4$$ (62)

$$4^4+21^4+22^4+26^4+28^4 = 35^4$$ (63)

$$1^4+ 2^4+12^4+24^4+44^4 = 45^4$$ (64)

$$1^4+ 8^4+12^4+32^4+64^4 = 65^4$$ (65)

$$2^4+39^4+44^4+46^4+52^4 = 65^4$$ (66)

$$22^4+52^4+57^4+74^4+76^4 = 95^4$$ (67)

$$22^4+28^4+63^4+72^4+94^4 = 105^4$$ (68)

(Berndt 1994). Berndt and Bhargava (1993) and Berndt (1994, pp. 94-96) give Ramanujan's solutions for arbitrary $s$, $t$, $m$, and $n$,

$(8s^2+40st-24t^2)^4+(6s^2-44st-18t^2)^4$
$+(14s^2-4st-42t^2)^4+(9s^2+27t^2)^4+(4s^2+12t^2)^4$
$=(15s^2+45t^2)^4,\quad$	(69)

and

$(4m^2-12n^2)^4+(3m^2+9n^2)^4+(2m^2-12mn-6n^2)^4$
$+(4m^2+12n^2)^4+(2m^2+12mn-6n^2)^4=(5m^2+15n^2)^4.$
	(70)

These are also given by Dickson (1966, p. 649), and two general Formulas are given by Beiler (1966, p. 290). Other solutions are given by Fauquembergue (1898), Haldeman (1904), and Martin (1910).

Ramanujan gave

$$2(ab+ac+bc)^2=a^4+b^4+c^4$$ (71)

$$2(ab+ac+bc)^4=a^4(b-c)^4+b^4(c-a)^4+c^4(a-b)^4$$ (72)

$$2(ab+ac+bc)^6=(a^2b+b^2c+c^2a)^4+(ab^2+bc^2+ca^2)^4+(3abc)^4$$ (73)

$2(ab+ac+bc)^8=(a^3+2abc)^4(b-c)^4$
$+(b^3+2abc)^4(c-a)^4+(c^3+2abc)^4(a-b)^4,\quad$	(74)

where

$$a+b+c=0$$ (75)

(Berndt 1994, pp. 96-97). Formula (72) is equivalent to Ferrari's Identity

$(a^2+2ac-2bc-b^2)^4+(b^2-2ab-2ac-c^2)^4$
$+(c^2+2ab+2bc-a^2)^4=2(a^2+b^2+c^2-ab+ac+bc)^4.\quad$	(76)

Bhargava's Theorem is a general identity which gives the above equations as a special case, and may have been the route by which Ramanujan proceeded. Another identity due to Ramanujan is

$$(a+b+c)^4+(b+c+d)^4+(a-d)^4 = (c+d+a)^4+(d+a+b)^4+(b-c)^4,$$ (77)

where $a/b=c/d$, and 4 may also be replaced by 2 (Ramanujan 1957, Hirschhorn 1998).

V. Kyrtatas noticed that $a=3$, $b=7$, $c=20$, $d=25$, $e=38$, and $f=39$ satisfy

$${a^4+b^4+c^4\over d^4+e^4+f^4}={a+b+c\over d+e+f}$$ (78)

and asks if there are any other distinct integer solutions.

The first few numbers $n$ which are a sum of two or more fourth Powers ($m-1$ equations) are 353, 651, 2487, 2501, 2829, ... (Sloane's A003294). The only number of the form

$$4x^4+y^4$$ (79)

which is Prime is 5 (Baudran 1885, Le Lionnais 1983).

See also Bhargava's Theorem, Ford's Theorem

References

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