1.Introduction

About the problem D16 in Unsolved Problems in Number Theory.(Guy's book[1])
D16: Triples with the same sum and the same product.

Schinzel[2] proved that for every k there exists k triples of positive integers
with the same sum and the same product.

Yong Zhang and Tianxin Cai[3] proved that for every k, there exists infinitely many primitive sets of k n-tuples of positive
integers with the same sum and the same product.

We show that for every k there exist k n-tuples of positive integers with the same sum and the same product.
Further, 6-tuple(distinct elements version) is shown.

2. Theorem

For every k there exists k n-tuples of positive integers with the same sum and the same product.

Proof.

We use Schinzel's system of equations below.

x+y+z = 6..........................................................(1)
xyz = 6............................................................(2)

Substitute z = 6-x-y to equation (1), we obtain

-yx^2+(-y^2+6y)x-6.................................................(3)

Since equation (3) is a quadratic equation in y, for y to be rational number, the discriminant of the equation must be square number.

Let U = x then we obtain

V^2 = U^4-12U^3+36U^2-24U..........................................(4)
    
Quartic equation (4) has a rational point Q(U,V)=(0,0), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2 = X^3-9X+9.....................................................(5)

Transformation is given, 

U = -6/(X-3)
V = -12Y/((X-3)^2)
X = (-6+3U)/U
Y = -3V/(U^2)

Elliptic curve (5) has rank 1 and generator P(X,Y)=(1,1).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

Substitute x = 6/(3-X) and V = -12Y/((X-3)^2) to equation (3), we obtain

y = (6-3X-Y)/(3-X)
z = (6-3X+Y)/(3-X).

Schinzel pointed out that if X is less than 2 then y and z are positive.

When X is rational point on elliptic curve (5) with X < 2, X is lying on egg.(see figure)

Hence taking odd multiples of any point on egg, we can obtain the solution such that y and z are positive.

Construction of n-tuples.
First, taking three rational points(P1,P2,P3) on egg and (Q1,Q2,Q3) that is corresponding points of quartic curve.
Thus we obtain triple, (x1,y1,z1),(x2,y2,z2), and (x3,y3,z3).
By adding n-3 any nutural number to triple, we can obtain n-tuple.
4-tuple: (x1,y1,z1,4),(x2,y2,z2,4), and (x3,y3,z3,4).
5-tuple: (x1,y1,z1,4,5),(x2,y2,z2,4,5), and (x3,y3,z3,4,5).
6-tuple: (x1,y1,z1,4,5,6),(x2,y2,z2,4,5,6), and (x3,y3,z3,4,5,6).
  .
  .
  .
In this way, we can construct n-tuple for any n.

Since there are infinitely many rational points on elliptic curve (5), then we can choose any k.


         Y^2=X^3-9X+9, |Y|=6-3X


Q.E.D.


3.Examples

We use 3 points such as {x1+y1+z1 = x2+y2+z2 = x3+y3+z3, x1y1z1 = x2y2z2 = x3y3z3} below.
Q1(x1,y1,z1)=(3, 1, 2)
Q2(x2,y2,z2)=(54/35, 25/21, 49/15)
Q3(x3,y3,z3)=(15123/16159, 25538/10153, 20449/8023)
Of course, we can use other rational points.

4-tuple:

[361396035, 120465345, 240930690, 481861380]
[185860818, 143411125, 393520127, 481861380]
[112741965, 303008370, 307041735, 481861380]

sum:1204653450, product:5054285343060869173236390226815000

5-tuple:

[361396035, 120465345, 240930690, 481861380, 602326725]
[185860818, 143411125, 393520127, 481861380, 602326725]
[112741965, 303008370, 307041735, 481861380, 602326725]

sum:1806980175, product:3044331137901354804768932576139486130875000

6-tuple(distinct elements version):

[1450743896263780614666445815048040135986504284982419136006330206901555, 483581298754593538222148605016013378662168094994139712002110068967185, 967162597509187076444297210032026757324336189988279424004220137934370, 733398742887922099355477705869307999150635260516891175312516037828985, 584057816606391615377111046806951198214058051717366463042461863939330, 1584031233033247514600302877419821074608315257730580633657682512034795]
[746096860935658601828457847738992069935916489419529841374684106406514, 575692022326897069312081672638111165074009636897785371431083415437125, 1579698909265005558192352109718977036963082443647523059206892891959471, 1438529267900403299868582637731350511125324359706189473180065091037250, 479707767015947730579275940751018766648234527886386162021365428453033, 983250757611210198885033051613710994199449682372262636811229894312827]
[452577509812842259950093035067588979856919865127568219853203203972445, 1216359618594977817306927122515409205582039673984077609091882885973010, 1232550664119741152075871472513082086534049030853192443067574323857655, 1462560335407809248413304369700276929961670367969305039346750421823335, 487720321850878693973330861740363417703416890305733883886824804850130, 951207135268873286946256398655439924307921311689799348779085187129645]

sum:5802975585055122458665783260192160543946017139929676544025320827606220
product:460383218393370750512338598957424318814313403462268164527884403642237299868766327686061594975406554803716588163725737290883127376238315469506766600887598665958525905954581357586649712377172271752741025171854958776250157377428996428944164867727346811873015760134194345435943305335182424099611664819855493694424796063373913911718782664338860184907876243561909361252812945593393353663540112501270946591837391430062500


4.References

[1] R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer Science, 2004.
[2] A. Schinzel, Triples of positive integers with the same sum and the same product, Serdica Math. J., 22(1996)
[3] Yong Zhang and Tianxin Cai, N-tuples of Positive Integers with the Same Sum and the Same Product, Math. Comput. 2012.