1,2

InputMode:=Word

[x+y=a, x^2+y^2=b, x^3+y^3=c]

"Find integer values of a, b and c."

; Simp(User)

(x+y)^2=a^2

; Expd(#4)

x^2+2*x*y+y^2=a^2

; Simp(User)

2*x*y=a^2-b

; Simp(User)

x*y=(a^2-b)/2

; Simp(User)

x^2-x*y+y^2=c/(x+y)

; Simp(Sub(User'))

-x*y=c/a-b

; Simp(User)

0=(a^3-3*a*b+2*c)/(2*a)

a^3-3*a*b+2*c=0

"Let d be a integer solution for a."

; Simp(User)

(a^3-3*a*b+2*c)/(a-d)=0

; Expd(#13)

(3*b*d-2*c-d^3)/(a-d)-a^2-a*d+3*b-d^2=0

"The other two solutions are solutions of these equations."

(3*b*d-2*c-d^3)/(a-d)=0

-a^2-a*d+3*b-d^2=0

; Simp(User)

c=d*(3*b-d^2)/2

; Simp(Solve(#17, a))

a=(SQRT(3)*SQRT(4*b-d^2)-d)/2 OR a=-(SQRT(3)*SQRT(4*b-d^2)+d)/2

LIST_TO_SET(v):=MAP_LIST(v SUB i, i, {1, ..., DIM(v)})

VECTOR(VECTOR(SUBST(a^3-3*a*b+2*c, [b, c], [x SUB 1, x SUB 2]), a, x SUB 3), x, a11)

"Let e be the second integer solution for a."

e=(SQRT(3)*SQRT(4*b-d^2)-d)/2

b=(d^2+d*e+e^2)/3

c=d*(3*b-d^2)/2

; Simp(User)

c=d*e*(d+e)/2

f=-(SQRT(3)*SQRT(4*b-d^2)+d)/2

; Simp(User)

f=-d-e

a9:=VECTOR(VECTOR([(d^2+d*e+e^2)/3, d*e*(d+e)/2, [d, e, -d-e]], e, d, 32), d, 1, 32)

a10:=SELECT(INTEGER?(x SUB 1) AND INTEGER?(x SUB 2), x, APPEND(a9))

a11:=SORT(a10)

; Simp(#31)

For values of c see A108940.

Sequence in context: A047537 A062458 A080902 this_sequence A082869 A166569 A024608

Adjacent sequences: A108284 A108285 A108286 this_sequence A108288 A108289 A108290

nonn

James R FitzSimons (cherry(AT)getnet.net), Jun 22 2007