D:\Final\...\4_ARMEN_33--41.DVI


Mathematical Problems of Computer Science 48, 33{41, 2017.

A N ew M ethod of Solving Diophantine E quation

a3 + b3 + c 3 = d

A r m e n A . A va g ya n

Armenian State Pedagogical University

e-mail: avagyana73@gmail.com

Abstract

The article is dedicated to the famous Diophantine equation of the form a3+b3+c3 =
d. We solve this problem in some particular cases. Using the package Mathematica
11 we ¯nd an e±cient algorithm to solve this problem. This algorithm is simpler and
uses a signi¯cantly smaller number of operations than the other known algorithms for
solving these equations.

Keywords: Diophantine equations, the sum of three cubes, Parametric solutions,
Polynomial identities.

1 . In t r o d u c t io n

Fe r m a t 's e qu a t io n fo r o d d e xp o n e n t s n a s ks fo r t h r e e in t e g e r s , e a c h wit h a n a b s o lu t e
va lu e g r e a t e r t h a n 0 , s u c h t h a t t h e s u m o f t h e ir n¡ t h p o we r s is z e r o . A r e la t e d p r o b le m
is t o ¯ n d t h r e e in t e g e r s , e a c h wit h a n a b s o lu t e va lu e g r e a t e r t h a n t h e n¡ t h r o o t o f k, s u c h
t h a t t h e s u m o f t h e ir n¡ t h p o we r s e qu a ls k.

Fo r e xa m p le , d e t e r m in e t h e in t e g e r s a; b; c; wit h jaj ; jbj ; jcj > 1 s u c h t h a t
a3 + b3 + c3 = d: ( 1 )

Th is h a s in ¯ n it e ly m a n y s o lu t io n s b e c a u s e o f t h e id e n t it y
³
1 ¡ 9 m3

´3
+

³
9 m4

´3
+

³
¡ 9 m4 + 3 m

´3
= 1 : ( 2 )

H o we ve r , t h e r e a r e o t h e r s o lu t io n s a s we ll. A r e t h e r e a n y o t h e r id e n t it ie s t h a t g ive a d i®e r e n t
1 -p a r a m e t e r fa m ily o f s o lu t io n s ?

Is e ve r y s o lu t io n o f ( 1 ) a m e m b e r o f a fa m ily like t h is ? In g e n e r a l, it is kn o wn t h a t t h e r e
is n o ¯ n it e m e t h o d fo r d e t e r m in in g wh e t h e r a g ive n D io p h a n t in e e qu a t io n h a s s o lu t io n s .
H o we ve r , it is a n o p e n p r o b le m , wh e t h e r t h e r e is a g e n e r a l m e t h o d fo r d e t e r m in in g if a g ive n
D io p h a n t in e e qu a t io n h a s " a lg e b r a ic " s o lu t io n s , i.e ., a n a lg e b r a ic id e n t it y like t h e o n e a b o ve
t h a t g ive s a n in ¯ n it e fa m ily o f s o lu t io n s . Mo r e s p e c ī c a lly, is t h e r e a p r o p o s it io n , t h a t o n ly
e qu a t io n s o f genus < 2 c a n h a ve a n a lg e b r a ic s o lu t io n ?

It m a y b e wo r t h m e n t io n in g t h a t t h e c o m p le t e r a t io n a l-s o lu t io n o f t h e e qu a t io n a3 + b3 +
c3 = t3 is kn o wn , a n d is g ive n b y

a = q
h
1 ¡ ( x ¡ 3 y )

³
x2 + 3 y2

´i
;

3 3



3 4 A New Method of Solving Diophantine Equation a3 + b3 + c3 = d

b = ¡q
h
1 ¡ ( x + 3 y )

³
x2 + 3 y2

´i
;

c = q
·³

x2 + 3 y2
´2

¡ ( x + 3 y )
¸
;

t = q
·³

x2 + 3 y2
´2

¡ ( x ¡ 3 y )
¸
;

wh e r e q; x; y a r e a n y r a t io n a l n u m b e r s .
Th u s , if we s e t q e qu a l t o t h e in ve r s e o fh

( x2 + 3 y2 )
2 ¡ ( x ¡ 3 y )

i
we h a ve r a t io n a l s o lu t io n s o f ( 1 ) . H o we ve r , I t h in k t h e p r o b le m o f

¯ n d in g t h e in t e g e r -s o lu t io n s is m o r e d i± c u lt . If t is a llo we d t o b e a n y in t e g e r ( n o t ju s t 1 )
t h e n R a m a n u ja n g a ve t h e in t e g e r s o lu t io n s a s

a = 3 n2 ¡ 5 n m ¡ 5 m2
b = 4 n2 ¡ 4 n m + 6 m2
c = 5 n2 ¡ 5 n m ¡ 3 m2
t = 6 n2 ¡ 4 n m ¡ 4 m2

Th is o c c a s io n a lly g ive s a s o lu t io n o f e qu a t io n ( 1 ) ( wit h a p p r o p r ia t e c h a n g e s in s ig n ) , a s in
t h e fo llo win g c a s e s
n m a b c
| - | { | | | | { | | {
1 -1 1 2 -2
1 -2 9 1 0 -1 2
5 -1 2 -1 3 5 -1 3 8 1 7 2
1 9 -8 -7 9 1 -8 1 2 1 0 1 0
4 6 -1 0 9 1 1 1 6 1 1 1 4 6 8 -1 4 2 5 8
7 3 -1 7 3 6 5 6 0 1 6 7 4 0 2 -8 3 8 0 2
4 1 9 -9 9 3 -9 5 1 6 9 0 -9 2 6 2 7 1 1 1 8 3 2 5 8

H o we ve r , t h is d o e s n o t c o ve r a ll t h e s o lu t io n s g ive n b y ( 2 ) . B y t h e wa y, t h e e qu a t io n
a3 +b3 +c3 = 1 h a s a lg e b r a ic s o lu t io n s [1 ], [2 ] o t h e r t h a n ( 2 ) . Th e r e a r e kn o wn t o b e in ¯ n it e ly
m a n y a lg e b r a ic s o lu t io n s , fo r in s t a n c e :

( 1 ¡ 9 t3 + 6 4 8 t6 + 3 8 8 8 t9 ) 3 + ( ¡1 3 5 t4 + 3 8 8 8 t10 ) 3 + ( 3 t ¡ 8 1 t4 ¡ 1 2 9 6 t7 ¡ 3 8 8 8 t10 ) 3 = 1

H o we ve r , it is n o t kn o wn wh e t h e r e ve r y s o lu t io n o f t h e e qu a t io n lie s in s o m e fa m ily o f
s o lu t io n s wit h a n a lg e b r a ic p a r a m e t e r iz a t io n .

In t e r e s t in g ly, n o t e , t h a t if yo u r e p la c e 1 b y 2 , t h e n a g a in t h e r e is a p a r a m e t r ic s o lu t io n :

³
6 t3 + 1

´3
¡

³
6 t3 ¡ 1

´3
+

³
6 t2

´3
= 2 ( 3 )

a n d a g a in t h is d o e s n o t c o ve r a ll t h e kn o wn in t e g e r s o lu t io n s . N o t e , t h a t p r e c is e ly o n e
s o lu t io n is kn o wn t h a t is n o t g ive n b y ( 3 ) ( s e e [1 ]) :

1 2 1 4 9 2 8 3 + 3 4 8 0 2 0 5 3 ¡ 3 5 2 8 8 7 5 3 = 2 .

It is e vid e n t ly n o t kn o wn u n t il t o d a y, if t h e r e a r e a n y o t h e r a lg e b r a ic s o lu t io n s b e s id e s t h e
o n e n o t e d a b o ve .

Fo r d > 2 K e n ji K o ya m a [3 ] h a s g e n e r a t e d a la r g e t a b le o f in t e g e r s o lu t io n s o f
a3 + b3 + c3 = d fo r n o n c u b e s d in t h e r a n g e 1 · d · 1 0 0 0 a n d jaj · jbj · jcj · 2 21 ¡ 1



A. Avagyan 3 5

c o n s is t s o f t wo t a b le s : Ta b le 1 ( 5 5 p a g e s ) c o n t a in s t h e in t e g e r s o lu t io n s , s o r t e d b y d, a n d
Ta b le 2 ( 2 p a g e s ) lis t s t h e n u m b e r o f p r im it ive s o lu t io n s fo u n d fo r e a c h d in t h e s e a r c h r a n g e .

In g e n e r a l, it s e e m s t o b e a d i± c u lt p r o b le m t o c h a r a c t e r iz e a ll t h e s o lu t io n s o f
a3 + b3 + c3 = d fo r s o m e a r b it r a r y in t e g e r d > 2 . In p a r t ic u la r , t h e qu e s t io n o f wh e t h e r a ll
in t e g e r s o lu t io n s a r e g ive n b y a n a lg e b r a ic id e n t it y s e e m s b o t h d i± c u lt a n d in t e r e s t in g .

N e ve r t h e le s s , fo r in s t a n c e , in t h e c a s e o f d = 3 , t h e r e is s t ill n o s o lu t io n kn o wn a p a r t fr o m
t h e o b vio u s o n e s : ( 1 , 1 , 1 ) , ( 4 , 4 ,5 ) , ( 4 ,5 , 4 ) , a n d ( 5 , 4 , 4 ) . Fo r d = 3 0 , t h e ¯ r s t s o lu t io n wa s
fo u n d b y N . E lkie s a n d h is c o wo r ke r s in 2 0 0 0 [5 ]. It is in t e r e s t in g , t h a t in 1 9 9 2 , D . R . H e a t h -
B r o wn [6 ] m a d e a p r e d ic t io n o n t h e d e n s it y o f t h e s o lu t io n s fo r d = 3 0 wit h o u t kn o win g a n y
s o lu t io n e xp lic it ly. Ove r t h e ye a r s , a n u m b e r o f a lg o r it h m s h a ve b e e n d e ve lo p e d in o r d e r
t o a t t a c k t h e g e n e r a l p r o b le m . Co n c e r n in g t h e va r io u s a p p r o a c h e s , a n e xc e lle n t o ve r vie w,
in ve n t e d b e fo r e 2 0 0 0 , wa s g ive n in [7 ], wh ic h wa s p u b lis h e d in 2 0 0 7 . H is t o r ic a lly, t h e ¯ r s t
a lg o r it h m , wh ic h h a s a c o m p le xit y o f O ( B1+² ) fo r a s e a r c h b o u n d o f B is t h e m e t h o d b y R .
H e a t h -B r o wn [8 ].

R e t u r n t h e in t e r e s t e d r e p r e s e n t a t io n s

d = a3 + b3 + c3 ( 4 )

o f va r io u s in t e g e r s d a s s u m s o f t h r e e c u b e s .
Th e c u b ic r e s id u e s wit h r e s p e c t t o m o d u le 9 a r e : 0 , 1 , 8 , t h u s , it fo llo ws b y in s p e c t io n o f

c a s e s t h a t fo r e ve r y in t e g e r s o lu t io n t o ( 4 ) we o b t a in d 6= 6 § 4 ( m o d 9 ) . A n y g ive n s o lu t io n
c a n b e wr it t e n in o n e o f t h e fo llo win g fo r m s fo r n o n -n e g a t ive a; b; c :

jdj = a3 + b3 + c3 o r jdj = a3 + b3 ¡ c3 o r jdj = a3 ¡ b3 ¡ c3

Th e r e fo r e , it s u ± c e s t o c o n s id e r n o n -n e g a t ive s o lu t io n s t o t h e e qu a t io n s a3 + b3 = c3 § d
a n d a3 + b3 + c3 = d ( fo r d = 0 it is a c a s e o f Fe r m a t s t h e o r e m t h a t t h e r e a r e n o in t e g e r
s o lu t io n s ) .

In p r a c t ic e , we n e e d t o s e a r c h fo r p r im it ive s o lu t io n s , i.e ., GCD ( a; b; c) is n o t d ivis ib le
b y d, s in c e t h e n o n -p r im it ive s o lu t io n s fo r ¯ xe d n a r e r o u t in e ly o b t a in e d fr o m t h e p r im it ive
s o lu t io n s fo r t h e ir d ivis o r s .

Co n s id e r in g d = m3; d = m12 a n d d = 2 m9 t yp e va lu e s o f d, a n d m u lt ip lyin g b o t h s id e s
o f ( 3 ) b y m9, a ft e r a p p lyin g t h e c h a n g e o f va r ia b le tt=m, o n e c a n o b t a in t h e m o r e g e n e r a l
s o lu t io n ³

6 t3 + m3
´3

¡
³
6 t3 ¡ m3

´3
¡

³
6 t2

´3
= 2 m9 ( 5 )

wh ic h is p r im it ive fo r GCD ( 6 t; m ) = 1 . If GCD ( 6 t; m) > 1 , t h e n d ivid in g ( 5 ) b y
( GCD ( 6 t3; m3 ) ) 3 g ive s a p r im it ive s o lu t io n . Fo r l; k ¸ 1 t h e s o lu t io n s a r e :

( 3 t3 + 2 3l¡1m3 ) 3 ¡ ( 3 t3 ¡ 2 3l¡1m3 ) 3 ¡ ( 2 l 3 mt2 ) 3 = 2 9l¡2m9 ( 6 )

( 2 t3 + 3 3k¡1m3 ) 3 ¡ ( 2 t3 ¡ 3 3k¡1m3 ) 3 ¡ ( 2 3 kmt2 ) 3 = 2 3 9k¡3m9 ( 7 )

( t3 + 2 3l¡1 3 3k¡1m3 ) 3 ¡ ( t3 ¡ 2 3l¡1 3 3k¡1m3 ) 3 ¡ ( 2 l 3 kmt2 ) 3 = 2 9l¡2 3 9k¡3m9 ( 8 )
N o t e , t h a t t h e y a r e p r im it ive fo r GCD ( 3 t; 2 m ) = 1 , GCD ( 2 t; 3 m ) = 1 a n d GCD ( t; 6 m) = 1 ,
r e s p e c t ive ly.

Th e la s t e qu a t io n s g ive p o lyn o m ia l fa m ilie s fo r n = 2 , 1 2 8 , 1 4 5 8 , 6 5 5 3 6 , 9 3 3 1 2 , 3 9 0 6 2 5 0 ,
2 8 6 9 7 8 1 4 , e t c .



3 6 A New Method of Solving Diophantine Equation a3 + b3 + c3 = d

A n a n a lo g o u s p r o c e d u r e m a y b e a p p lie d fo r 3 t o o b t a in fa m ilie s o f s o lu t io n s fo r n u m b e r s
o f t h e fo r m m12. Mu lt ip lyin g b o t h s id e s b y m12 a n d a p p lyin g t h e t r a n s fo r m a t io n t=m, we
will g e t ³

9 mt3 + m4
´3

¡
³
9 t4 + 3 mt

´3
+

³
9 t4

´3
= m12 ( 9 )

wh ic h is p r im it ive fo r GCD ( t; 3 m ) = 1 . In p a r t ic u la r , fo r 3 ¡ m a n d k 1 ,
³
3 kmt3 + 3 4k¡2m4

´3
¡

³
t4 + 3 3k¡1lm3t

´3
+

³
t4

´3
= 3 12k¡6m12 ( 1 0 )

is p r im it ive fo r GCD ( t; 3 m ) = 1 . E qu a t io n s ( 9 ) a n d ( 1 0 ) g ive fa m ilie s o f s o lu t io n s fo r n =
1 , 7 2 9 , 4 0 9 6 , 2 9 8 5 9 8 4 , 1 6 7 7 7 2 1 6 , 2 4 4 1 4 0 6 2 5 , 3 8 7 4 2 0 4 8 9 ,e t c .

2 . N e w Me t h o d a n d R e s u lt s

Co n s id e r in g t h e m o r e g e n e r a liz e d p r o b le m o f t h e s u m o f t h r e e c u b e s , we a r e s e e kin g
P1 ( y ) ; P2 ( y ) ; P3 ( y ) wit h t h e h ig h e s t p o s s ib le d e g r e e p o lyn o m ia ls a n d Q ( y ) wit h t h e lo we s t
p o s s ib le d e g r e e p o lyn o m ia l, s u c h a s

P 31 ( y ) + P
3
2 ( y ) + P

3
3 ( y ) = Q ( y )

A c t u a lly, t h e s o lu t io n o f t h is p r o b le m h a s a c lo s e r e la t io n wit h t h e a b o ve t r ivia l p r o b le m ,
s in c e t h e c a s e o f degQ ( y ) = 0 c o in c id e s wit h o u r p r o b le m . N e ve r t h e le s s , t h e e s t im a t io n o f
p o s s ib ilit y o f m in im iz a t io n o f degQ( y ) it s e lf is a ls o a n in t e r e s t in g p r o b le m .

Result 1: Th e ¯ r s t r e s u lt o f t h is p a p e r is d e vo t e d t o t h e c a s e o f d e g r e e s ( 8 , 8 , 6 ) . W e
s e a r c h t h e d e s ir e d p o lyn o m ia ls wit h in t h e c la s s o f p o lyn o m ia ls o f t h e fo r m

³
ax8 + bx5 + cx2

´3
¡

³
ax8 + b1x

5 + c1x
2
´3

¡
³
Ax6 + Bx3 + C

´3
( 1 1 )

Fir s t o f a ll, we e xp a n d it

¡C3 ¡ 3 BC2x3 +
³
c3 ¡ 3 B2C ¡ 3 AC2 ¡ c31

´
x6 +

³
¡B3 + 3 bc2 ¡ 6 ABC ¡ 3 b1c21

´
x9

+
³
¡ 3 AB2 + 3 b2c + 3 ac2 ¡ 3 A2C ¡ 3 b21c1 ¡ 3 ac21

´
x12+

³
b3 ¡ 3 A2B ¡ b31 + 6 abc ¡ 6 ab1c1

´
x15

+
³
¡A3 + 3 ab2 ¡ 3 ab21 + 3 a2c ¡ 3 a2c1

´
x18 +

³
3 a2b ¡ 3 a2b1

´
x21

Fu r t h e r ,we t a ke b1 = b, c1 =
¡A3+3a2c

3a2
, B = 2Ab

3a
, C = ¡A

4¡aAb2+6a2Ac
9a3

a n d g e t t h e fo llo win g
fo r m :

A12

7 2 9 a9
+

A9b2

2 4 3 a8
+

A6b4

2 4 3 a7
+

A3b6

7 2 9 a6
¡ 2 A

9c

8 1 a7
¡ 4 A

6b2c

8 1 a6
¡ 2 A

3b4c

8 1 a5
+

4 A6c2

2 7 a5
+

4 A3b2c2

2 7 a4
¡ 8 A

3c3

2 7 a3
+

+

Ã
¡ 2 A

9b

8 1 a7
¡ 4 A

6b3

8 1 a6
¡ 2 A

3b5

8 1 a5
+

8 A6bc

2 7 a5
+

8 A3b3c

2 7 a4
¡ 8 A

3bc2

9 a3

!
x3+

+

Ã
2 A6b2

2 7 a5
+

A3b4

9 a4
+

A6c

9 a4
¡ 4 A

3b2c

9 a3
¡ A

3c2

3 a2

!
x6 +

Ã
A6b

9 a4
+

4 A3b3

2 7 a3
¡ 2 A

3bc

3 a2

!
x9

Th u s , fu r t h e r c o n s id e r a t io n s a r e d e vo t e d t o ¯ n d in g t h e c a s e s , wh ic h a r e in t e r e s t in g fo r u s :



A. Avagyan 3 7

Case 1: b = 0 . Th e r e s u lt h a s t h e fo r m

A12

7 2 9 a9
¡ 2 A

9c

8 1 a7
+

4 A6c2

2 7 a5
¡ 8 A

3c3

2 7 a3
+

Ã
A6c

9 a4
¡ A

3c2

3 a2

!
x6

Subcase 1.1: c = 0 . Th e r e s u lt g e t s t h e fo r m A
12

729a9
, wh ic h is a c u b e o f a n in t e g e r , t h u s ,

it is p r im it ive a n d n o t in t e r e s t in g .
Subcase 1.2: c = A

3

3a2
, t h e r e s u lt is ¡ A12

729a9
, a g a in p r im it ive .

Case 2: c = 3A
3+4ab2

18a2
. Th e r e s u lt :

¡ A
3b6

1 9 6 8 3 a6
¡ 2 A

3b5x3

7 2 9 a5
+

Ã
A9

1 0 8 a6
¡ A

3b4

2 4 3 a4

!
x6

Fa c t o r iz in g t h e c o e ± c ie n t o f t h e la s t t e r m , o n e c a n o b t a in :

¡A
3 ( ¡ 3 A3 + 2 ab2 ) ( 3 A3 + 2 ab2 )

9 7 2 a6

S u b s t it u t in g a = 3A
3

2b2
. Th e n t h e r e s u lt will g e t t h e fo r m :

¡ 6 4 b
18

1 4 3 4 8 9 0 7 A6
¡ 6 4 b

15x3

1 7 7 1 4 7 A3

Fin a lly, t a kin g x = 2yb
A

we o b t a in t h e r e s u lt , wh ic h is in t e r e s t in g fo r u s : ¡1 ¡ 6 4 8 y3.
P r actical consider ations: N o w we in ve s t ig a t e t h is r e s u lt fo r a p p lic a t io n s t o s o lve t h e

p r o c e s s o f t h e e qu a t io n ( 4 ) . S in c e fo r max[abs[a; b; c]] · 1 0 14 t h e r e a r e we ll-kn o wn t a b le s
in [4 ], t h u s , we s e e k s o lu t io n s o f ( 4 ) s a t is fyin g t h e c o n d it io n max[abs[a; b; c]] ¸ 1 0 15 wit h
p o s s ib le s m a ll va lu e s o f abs[d] ( d e s ir a b ly le s s t h a n 1 0 0 0 ) .

Th u s , t h e r e s u lt is :

³
5 4 y2

³
1 + 3 6 y3 + 4 3 2 y6

´´3
¡

³
1 8 y2

³
1 + 1 0 8 y3 + 1 2 9 6 y6

´´3
¡

³
1 + 2 1 6 y3 + 3 8 8 8 y6

´3

= ¡ 1 ¡ 6 4 8 y3

S u r e , c a lc u la t io n s we r e e xp e c t e d t o b e s ig n i¯ c a n t ly h a r d , fo r wh ic h we will u s e Ma t h e -
m a t ic a 1 1 .0 c o d e :

G8[y ] := 1
.

GCD
h³

5 4 y2
³
1 + 3 6 y3 + 4 3 2 y6

´´
;
³
1 8 y2

³
1 + 1 0 8 y3 + 1 2 9 6 y6

´´
;

³
1 + 2 1 6 y3 + 3 8 8 8 y6

´i

F8[y ]:=G8[y]
n³

5 4 y2
³
1 + 3 6 y3 + 4 3 2 y6

´´
;
³
1 8 y2

³
1 + 1 0 8 y3 + 1 2 9 6 y6

´´
;
³
1 + 2 1 6 y3 + 3 8 8 8 y6

´o

V8[y ] := G8[y]
3

³
¡1 ¡ 6 4 8 y3

´

F or[i = ¡ 5 0 ; i · 5 0 ; i + +
If[Abs[V8[i]] < 1 0 0 0 0 0 0 ; If[Max[Abs[F8[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 ; P rint[fi; F8[i]; V8[i]g]]]]

Th e r e s u lt is :

f¡ 1 1 ; f 5 0 0 0 2 5 0 8 9 9 3 5 8 ; 5 0 0 0 2 5 0 8 9 5 0 0 2 ; 6 8 8 7 5 4 1 6 7 3 g; 8 6 2 4 8 7 g



3 8 A New Method of Solving Diophantine Equation a3 + b3 + c3 = d

f¡ 1 0 ; f 2 3 3 2 6 0 5 6 0 5 4 0 0 ; 2 3 3 2 6 0 5 6 0 1 8 0 0 ; 3 8 8 7 7 8 4 0 0 1 g; 6 4 7 9 9 9 g
f¡ 9 ; f 1 0 0 4 0 7 9 1 2 0 6 0 6 ; 1 0 0 4 0 7 9 1 1 7 6 9 0 ; 2 0 6 6 0 8 5 1 4 5 g; 4 7 2 3 9 1 g
f 9 ; f 1 0 0 4 3 0 8 7 0 3 1 1 8 ; 1 0 0 4 3 0 8 7 0 0 2 0 2 ; 2 0 6 6 4 0 0 0 7 3 g; ¡4 7 2 3 9 3 g
f 1 0 ; f 2 3 3 2 9 9 4 4 0 5 4 0 0 ; 2 3 3 2 9 9 4 4 0 1 8 0 0 ; 3 8 8 8 2 1 6 0 0 1 g; ¡6 4 8 0 0 1 g
f 1 1 ; f 5 0 0 0 8 7 7 0 6 5 6 4 6 ; 5 0 0 0 8 7 7 0 6 1 2 9 0 ; 6 8 8 8 1 1 6 6 6 5 g; ¡8 6 2 4 8 9 g

Th is m e a n s t h a t , fo r in s t a n c e :

1 0 0 4 0 7 9 1 2 0 6 0 6 3 ¡ 1 0 0 4 0 7 9 1 1 7 6 9 0 3 ¡ 2 0 6 6 0 8 5 1 4 5 3 = 4 7 2 3 9 1 :

Result 2: Mo r e e n h a n c e d r e s u lt is o b t a in e d fo r t h e c a s e ( 9 , 9 , 7 ) :

( 3 + 3 6 0 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) 3 ¡ ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) 3

¡( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) 3 = 2 8 + 1 0 7 2 y3

G9[y¡] := 1 =GCD[( 3 + 3 6 0 y
3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ; ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ;

( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) ]

F9[y¡] := G9[y] ¤ f ( 3 ( 1 + 1 2 0 y3 + 3 4 5 6 y6 + 3 1 1 0 4 y9 ) ) ; ( ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ) ;
( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) g

V9[y¡] := G9[y]
3 ¤ ( 2 8 + 1 0 7 2 y3 )

F or[i = ¡ 5 0 ; i · 5 0 ; i + +;
If[Abs[V9[i]] < 1 0 0 0 0 0 0 0 ; If[Max[Abs[F9[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; P rint[fi; F9[i]; V9[i]g]]]

Th e r e s u lt is :

f¡ 2 1 ; f¡ 7 4 1 1 4 9 7 0 4 8 6 7 5 7 7 0 1 ; ¡ 7 4 1 1 4 9 7 0 4 8 5 4 2 4 1 2 1 ; ¡ 2 8 0 1 0 2 7 6 9 4 2 6 7 6 g; ¡ 9 9 2 7 7 6 4 g

f¡ 2 0 ; f¡ 4 7 7 7 5 0 8 0 4 5 0 8 7 9 9 9 7 ; ¡ 4 7 7 7 5 0 8 0 4 4 9 7 2 8 0 0 1 ; ¡ 1 9 9 0 6 3 5 2 6 4 0 4 0 0 g; ¡ 8 5 7 5 9 7 2 g
f¡ 1 9 ; f¡ 3 0 1 1 0 1 4 6 6 8 5 9 2 9 0 7 7 ; ¡ 3 0 1 1 0 1 4 6 6 8 4 9 4 1 3 8 5 ; ¡ 1 3 9 0 1 3 2 4 3 8 9 2 9 2 g; ¡ 7 3 5 2 8 2 0 g
f¡ 1 8 ; f¡ 1 8 5 0 8 9 4 9 4 6 6 1 7 9 9 0 1 ; ¡ 1 8 5 0 8 9 4 9 4 6 5 3 4 0 0 9 7 ; ¡9 5 2 1 1 0 9 8 8 9 1 2 8 g; ¡ 6 2 5 1 8 7 6 g
f¡ 1 7 ; f¡ 1 1 0 6 5 4 2 1 6 7 5 1 4 1 3 4 9 ; ¡ 1 1 0 6 5 4 2 1 6 7 4 4 3 3 8 8 1 ; ¡6 3 8 1 4 7 8 7 9 9 6 2 0 g; ¡ 5 2 6 6 7 0 8 g
f¡ 1 6 ; f¡6 4 1 2 1 7 7 8 6 8 4 8 8 7 0 1 ; ¡ 6 4 1 2 1 7 7 8 6 7 8 9 8 8 8 1 ; ¡ 4 1 7 4 6 2 3 2 7 7 3 7 6 g; ¡ 4 3 9 0 8 8 4 g
f¡ 1 5 ; f¡3 5 8 7 1 0 8 6 5 3 2 1 4 9 9 7 ; ¡ 3 5 8 7 1 0 8 6 5 2 7 2 9 0 0 1 ; ¡ 2 6 5 7 1 3 9 3 9 0 3 0 0 g; ¡ 3 6 1 7 9 7 2 g
f¡ 1 4 ; f¡1 9 2 7 8 4 5 5 3 2 2 6 7 1 9 7 ; ¡ 1 9 2 7 8 4 5 5 3 1 8 7 2 0 6 5 ; ¡ 1 6 3 9 3 4 1 0 2 7 3 5 2 g; ¡ 2 9 4 1 5 4 0 g

f 1 4 ; f 1 9 2 8 0 0 1 6 6 4 7 2 5 6 9 9 ; 1 9 2 8 0 0 1 6 6 4 3 3 0 5 5 9 ; 1 6 3 9 4 4 0 6 0 1 6 2 4 g; 2 9 4 1 5 9 6 g
f 1 5 ; f 3 5 8 7 3 4 4 8 4 9 2 1 5 0 0 3 ; 3 5 8 7 3 4 4 8 4 8 7 2 8 9 9 9 ; 2 6 5 7 2 7 0 6 1 0 3 0 0 g; 3 6 1 8 0 2 8 g
f 1 6 ; f 6 4 1 2 5 2 5 7 6 0 8 3 9 6 8 3 ; 6 4 1 2 5 2 5 7 6 0 2 4 9 8 5 5 ; 4 1 7 4 7 9 3 1 4 6 6 8 8 g; 4 3 9 0 9 4 0 g
f 1 7 ; f 1 1 0 6 5 9 2 2 1 9 1 7 7 2 1 3 9 ; 1 1 0 6 5 9 2 2 1 9 1 0 6 4 6 6 3 ; 6 3 8 1 6 9 5 2 8 6 0 5 2 g; 5 2 6 6 7 6 4 g
f 1 8 ; f 1 8 5 0 9 6 5 4 7 4 3 6 5 6 7 7 1 ; 1 8 5 0 9 6 5 4 7 4 2 8 1 6 9 5 9 ; 9 5 2 1 3 8 1 9 8 6 9 2 0 g; 6 2 5 1 9 3 2 g
f 1 9 ; f 3 0 1 1 1 1 2 2 2 2 9 3 1 7 4 9 9 ; 3 0 1 1 1 1 2 2 2 2 8 3 2 9 7 9 9 ; 1 3 9 0 1 6 6 2 1 8 1 3 2 4 g; 7 3 5 2 8 7 6 g



A. Avagyan 3 9

f 2 0 ; f 4 7 7 7 6 4 0 7 5 5 4 8 8 0 0 0 3 ; 4 7 7 7 6 4 0 7 5 5 3 7 2 7 9 9 9 ; 1 9 9 0 6 7 6 7 3 6 0 4 0 0 g; 8 5 7 6 0 2 8 g
f 2 1 ; f 7 4 1 1 6 7 4 8 9 3 3 0 4 2 7 6 3 ; 7 4 1 1 6 7 4 8 9 3 1 7 0 9 1 7 5 ; 2 8 0 1 0 7 8 1 0 3 7 4 2 8 g; 9 9 2 7 8 2 0 g

H e r e t h e m o s t in t e r e s t in g t r ip le t is :

1 9 2 8 0 0 1 6 6 4 7 2 5 6 9 9 3 ¡ 1 9 2 8 0 0 1 6 6 4 3 3 0 5 5 9 3 ¡ 1 6 3 9 4 4 0 6 0 1 6 2 4 3 = 2 9 4 1 5 9 6

Result 3: N o w, c o n s id e r in g t h e c a s e wh e n ( 2 5 ; 2 5 ; 1 8 ) , u s in g t h e s a m e a p p r o a c h we o b t a in :

µ
1

1 8
y ( 6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 )

¶3

¡
µ

1

1 8
y ( 6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 )

¶3
¡
µ

2

3
( 3 + 2 0 y6 + 3 2 y12 + 3 2 y18 )

¶3

= ¡ 8 ¡ 1 3 y6

Th u s , o n e u s e s t h e fo llo win g c o d e :

G25[y¡] := 1 =GCD[
µ

1

1 8
y

³
6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24

´¶
;

µ
1

1 8
y

³
6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24

´¶
;
µ

2

3

³
3 + 2 0 y6 + 3 2 y12 + 3 2 y18

´¶
]

F25[y¡] := G25[y] ¤ f
µ

1

1 8
y

³
6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24

´¶
;

µ
1

1 8
y

³
6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24

´¶
;
µ

2

3

³
3 + 2 0 y6 + 3 2 y12 + 3 2 y18

´¶
g

V25[y¡] := G25[y]
3 ¤ ( ¡8 ¡ 1 3 y6 )

F or[i = ¡ 5 0 ; i · 1 3 ; i + +;
If[Abs[V25[i]] < 1 0 0 0 0 0 0 0 0 0 ; If[Max[Abs[F25[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;

P rint[fi; F25[i]; V25[i]g]]]]
Th e r e s u lt is :

f¡ 2 8 ; f¡2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 3 0 7 9 6 0 1 ;
¡ 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 4 3 0 8 9 1 3 ; 1 1 9 3 6 3 9 7 9 2 9 6 4 3 8 8 5 1 9 0 1 0 2 2 2 0 8 1 g; ¡ 7 8 3 0 7 1 7 4 5 g
f¡ 2 4 ; f¡ 4 5 5 2 4 8 4 8 2 0 7 1 5 5 3 6 3 5 5 8 6 9 3 8 7 6 1 9 4 3 6 4 2 1 5 4 ; ¡ 4 5 5 2 4 8 4 8 2 0 7 1 5 5 3 6 3 5 5 8 6 9 3 8 7 6 1 9 4 4 3 0 5 7 0 6 ;

7 4 4 4 4 2 3 5 9 0 5 1 1 7 1 0 8 6 2 3 6 3 8 5 2 9 g; ¡ 3 1 0 5 4 2 3 3 7 g
f¡ 2 0 ; f¡ 4 7 7 2 1 8 5 9 9 6 2 9 2 5 5 2 6 4 0 2 8 4 4 5 4 3 9 9 8 4 0 0 3 5 ; ¡ 4 7 7 2 1 8 5 9 9 6 2 9 2 5 5 2 6 4 0 2 8 4 4 5 4 4 0 0 1 6 0 0 3 5 ;

2 7 9 6 2 0 2 7 1 0 3 5 7 3 3 3 7 6 0 0 0 0 0 0 1 g; ¡ 1 0 4 0 0 0 0 0 1 g
f¡1 8 ; f¡ 6 8 5 1 8 8 5 5 8 8 8 4 8 6 8 2 6 6 8 6 7 5 8 4 5 4 6 8 1 2 4 4 7 ; ¡ 6 8 5 1 8 8 5 5 8 8 8 4 8 6 8 2 6 6 8 6 7 5 8 4 5 4 7 2 3 2 3 5 1 ;

8 3 9 3 9 0 0 6 3 6 1 8 7 2 9 3 9 2 1 9 7 1 2 2 g; ¡ 4 4 2 1 5 8 9 2 0 g
f¡ 1 6 ; f¡ 1 8 0 2 8 8 1 0 1 4 8 4 8 0 4 3 9 4 8 5 7 6 4 9 2 1 6 5 5 3 2 4 ; ¡ 1 8 0 2 8 8 1 0 1 4 8 4 8 0 4 3 9 4 8 5 7 6 4 9 2 1 7 8 6 3 9 6 ;

5 0 3 7 1 9 1 2 1 5 3 0 0 9 4 1 2 3 7 4 5 2 9 g; ¡ 2 7 2 6 2 9 7 7 g
f¡ 1 4 ; f¡1 2 7 9 9 6 6 0 0 2 3 5 5 3 5 2 2 7 1 3 1 9 7 3 3 0 1 4 0 3 3 ; ¡ 1 2 7 9 9 6 6 0 0 2 3 5 5 3 5 2 2 7 1 3 1 9 7 3 3 1 6 7 6 9 7 ;



4 0 A New Method of Solving Diophantine Equation a3 + b3 + c3 = d

9 1 0 6 7 5 0 0 9 9 2 9 7 1 4 8 2 4 3 4 5 8 g; ¡9 7 8 8 3 9 7 6 g
f¡1 3 ; f¡ 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 7 8 0 8 2 3 9 ; ¡ 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 8 0 3 6 7 2 7 ;

4 7 9 8 0 9 8 3 5 7 3 3 5 2 7 6 0 4 7 6 0 4 g; ¡ 5 0 1 9 8 8 2 0 0 g
f¡ 1 2 ; f¡ 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 4 2 0 3 7 ; ¡ 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 8 3 5 0 9 ;

2 8 3 9 8 2 3 1 6 7 6 7 8 6 7 2 8 9 6 0 1 g; ¡4 8 5 2 2 2 5 g
f¡ 1 1 ; f¡6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 6 9 3 6 8 1 ; ¡ 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 8 1 0 8 0 9 ;

2 3 7 2 2 3 2 7 2 6 1 5 3 2 6 5 2 4 9 1 6 g; ¡ 1 8 4 2 4 2 4 0 8 g
f¡1 0 ; f¡ 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 5 9 9 9 8 0 0 3 5 ; ¡ 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 6 0 0 0 2 0 0 3 5 ;

2 1 3 3 3 3 5 4 6 6 6 6 8 0 0 0 0 0 0 2 g; ¡ 1 3 0 0 0 0 0 8 g
f¡ 9 ; f¡ 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 6 5 8 2 9 1 ; ¡ 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 7 1 0 7 7 9 ;

6 4 0 4 0 4 9 8 2 3 0 1 3 0 2 4 1 1 6 g; ¡ 5 5 2 6 9 9 2 8 g
f¡ 8 ; f¡ 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 8 3 2 4 6 ; ¡ 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 9 1 4 3 8 ;

1 9 2 1 5 4 3 1 7 1 1 0 6 4 0 6 4 1 g; ¡ 4 2 5 9 8 5 g
f¡ 7 ; f¡ 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 6 5 1 7 3 ; ¡ 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 8 4 3 8 1 ;

6 9 4 7 9 5 7 0 7 4 2 2 3 7 0 4 4 g; ¡1 2 2 3 5 5 6 0 g
f¡ 6 ; f¡ 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 3 3 3 3 ; ¡ 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 8 5 1 7 ;

2 1 6 6 6 5 8 8 4 7 5 7 1 4 5 8 g; ¡ 6 0 6 5 3 6 g
f 6 ; f8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 8 5 1 7 ; 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 3 3 3 3 ;

2 1 6 6 6 5 8 8 4 7 5 7 1 4 5 8 g; ¡ 6 0 6 5 3 6 g
f 7 ; f 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 8 4 3 8 1 ; 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 6 5 1 7 3 ;

6 9 4 7 9 5 7 0 7 4 2 2 3 7 0 4 4 g; ¡1 2 2 3 5 5 6 0 g
f 8 ; f 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 9 1 4 3 8 ; 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 8 3 2 4 6 ;

1 9 2 1 5 4 3 1 7 1 1 0 6 4 0 6 4 1 g; ¡ 4 2 5 9 8 5 g
f 9 ; f 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 7 1 0 7 7 9 ; 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 6 5 8 2 9 1 ;

6 4 0 4 0 4 9 8 2 3 0 1 3 0 2 4 1 1 6 g; ¡ 5 5 2 6 9 9 2 8 g
f 1 0 ; f 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 6 0 0 0 2 0 0 3 5 ; 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 5 9 9 9 8 0 0 3 5 ;

2 1 3 3 3 3 5 4 6 6 6 6 8 0 0 0 0 0 0 2 g; ¡ 1 3 0 0 0 0 0 8 g
f1 1 ; f 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 8 1 0 8 0 9 ; 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 6 9 3 6 8 1 ;

2 3 7 2 2 3 2 7 2 6 1 5 3 2 6 5 2 4 9 1 6 g; ¡ 1 8 4 2 4 2 4 0 8 g
f 1 2 ; f 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 8 3 5 0 9 ; 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 4 2 0 3 7 ;

2 8 3 9 8 2 3 1 6 7 6 7 8 6 7 2 8 9 6 0 1 g; ¡4 8 5 2 2 2 5 g
f 1 3 ; f 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 8 0 3 6 7 2 7 ; 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 7 8 0 8 2 3 9 ;

4 7 9 8 0 9 8 3 5 7 3 3 5 2 7 6 0 4 7 6 0 4 g; ¡ 5 0 1 9 8 8 2 0 0 g
H e r e t h e m o s t in t e r e s t in g t r ip le is :

( ¡ 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 3 0 7 9 6 0 1 ) 3+( 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 4 3 0 8 9 1 3 ) 3

¡( 1 1 9 3 6 3 9 7 9 2 9 6 4 3 8 8 5 1 9 0 1 0 2 2 2 0 8 1 ) 3 = ¡7 8 3 0 7 1 7 4 5



A. Avagyan 4 1

Refer ences

[1 ] G. P a yn e a n d L . N . V a s e r s t e in , \ S u m s o f t h r e e c u b e s " , The Arithmetic of F unction
F ields, d e Gr u yt e r , p p . 4 4 3 { 4 5 4 1 9 9 2 .

[2 ] D . H .L e h m e r , \ On t h e D io p h a n t in e e qu a t io n " ,J . L ondon M ath. Soc., vo l. 3 1 , p p . 2 7 5 { 2 8 0 ,
1 9 5 6 .

[3 ] K . K o ya m a , \ Ta b le s o f s o lu t io n s o f t h e D io p h a n t in e e qu a t io n " , M athematics of Compu-
tation, vo l. 6 2 , p p . 9 4 1 { 9 4 2 , 1 9 9 4 .

[4 ] D . B e r n s t e in , \ Th r e e c u b e s " , On lin e . A va ila b le : h t t p :/ / c r .yp .t o / t h r e e c u b e s .h t m l.

[5 ] N . D .E lkie s , \ R a t io n a l p o in t s n e a r c u r ve s a n d s m a ll n o n z e r o jx3y2j via la t t ic e r e d u c t io n " ,
Algorithmic number theory (L eiden 2000), L e c t u r e N o t e s in Co m p u t e r S c ie n c e 1 8 3 8 ,
S p r in g e r , B e r lin , p p . 3 3 { 6 3 , 2 0 0 0 .

[6 ] D . R . H e a t h -B r o wn , \ Th e d e n s it y o f z e r o s o f fo r m s fo r wh ic h we a k a p p r o xim a t io n fa ils " ,
M ath. Comp., vo l. 5 9 , p p . 6 1 3 { 6 2 3 , 1 9 9 2 .

[7 ] M. B e c k, E . P in e , W . Ta r r a n t a n d K . Y a r b r o u g h Je n s e n , \ N e w in t e g e r r e p r e s e n t a t io n s
a s t h e s u m o f t h r e e c u b e s " , M ath. Comp., vo l. 7 6 , p p . 1 6 8 3 { 1 6 9 0 , 2 0 0 7 .

[8 ] D . R . H e a t h -B r o wn , W . M. L io e n a n d H . J. J. t e R ie le , \ On s o lvin g t h e D io p h a n t in e
e qu a t io n o n a ve c t o r c o m p u t e r " , M ath. Comp., vo l. 6 1 , p p . 2 3 5 -2 4 4 , 1 9 9 3 .

Submitted 12.04.2017, accepted 10.11.2017.

a3 + b3 + c3 = d ï»ëùÇ ¹Çáý³ÝïÛ³Ý Ñ³í³ë³ñÙ³Ý ÉáõÍÙ³Ý Ýáñ Ù»Ãá¹

². ²í³·Û³Ý

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Ðá¹í³ÍÁ ÝíÇñí³Í ¿ ѳÛïÝÇ a3 + b3 + c3 = dï»ëùÇ ¹Çáý³ÝïÛ³Ý Ñ³í³ë³ñáõÙÝ»ñÇ
áñáß Ù³ëݳíáñ ¹»åù»ñÇ ÉáõÍÙ³ÝÁ:ÎÇñ³é»Éáí “Mathematica 11” ÷³Ã»ÃÁ, ѳçáÕí»É ¿
·ïÝ»É Ýßí³Í ËݹñÇ ÉáõÍÙ³Ý ³ñ¹Ûáõݳí»ï ³É·áñÇÃÙ, áñÝ ³Ûë ËݹñÇ ÉáõÍÙ³Ý ³ÛÉ
³É·áñÃÙÝ»ñÇ Ñ³Ù»Ù³ï ³í»ÉÇ å³ñ½ ¿ ¨ û·ï³·áñÍáõÙ ¿ ½·³ÉÇáñ»Ý ÷áùñ ù³Ý³ÏáõÃÛ³Ùµ
·áñÍáÕáõÃÛáõÝÝ»ñ:

Íîâûé ìåòîä ðåøåíèÿ Äèîôàíòîãî óðàâíåíèÿ a3 + b3 + c3 = d
À. Àâàãÿí

Àííîòàöèÿ

Ñòàòüÿ ïîñâÿùåíà ðåøåíèþ èçâåñòíîãî äèîôàíòîâîãî óðàâíåíèÿ âèäà a3 +b3 +
c3 = d â íåêîòîðûõ ÷àñòíûõ ñëó÷àÿõ. Èñïîëüçóÿ ïàêåò ”Mathematica 11” óäàëîñü
íàéòè ýôôåêòèâíûé àëãîðèòì íàõîæäåíèÿ ðåøåíèé ýòîé çàäà÷è, êîòîðûé
ïî ñðàâíåíèþ ñ äðóãèìè èçâåñòíûìè àëãîðèòìàìè, äàþùèìè ðåøåíèÿ ýòîé
çàäà÷è, ÿâëÿåòñÿ áîëåå ïðîñòûì è èñïîëüçóåò çíà÷èòåëüíî ìåíüøåå êîëè÷åñòâî
îïåðàöèé.