Identities Of Equal Sums Of Like Power

Identities of equal sums of like power can be sorted into the following categories:

- Constants
- Linear Forms
- Other equal sums of like power
- Miscellaneous

These identities were collected by Edward Brisse

Constants

3rd degree

x₀=9*a⁴y₀=-9*a⁴+3*a
z₀=-9*a³+1

x₁=9*a⁴
y₁=-9*a⁴-3*a
z₁=9*a³+1

For n= 2, 3, ...
x_n=(432*a⁶-2) * x_n-1 - x_n-2 - 108 * a⁴
y_n=(432*a⁶-2) * y_n-1 - y_n-2 - 108 * a⁴
z_n=(432*a⁶-2) * z_n-1 - z_n-2 + 216 * a⁶ + 4
x_n³ + y_n³ + z_n³ = 1 for n = 0, 1, ...

x₂=3888*a¹⁰-135*a⁴
y₂=-3888*a¹⁰-1296*a⁷-81a⁴+3*a
z₂=3888*a⁹+648*a⁶-9*a³+1

x₃=1679616*a¹⁶-66096*a¹⁰+153*a⁴
y₃=-1679616*a¹⁶-559872*a¹³-27216*a¹⁰+3888*a⁷+63*a⁴-3*a
z₃=1679616*a¹⁵+279936*a¹²-11664*a⁹-648*a⁶+9*a³+1
... ref03
64³+94³-103³ = 1 is not related to the previous formula

(6*a³+1)³+(-6*a³+1)³+(-6*a²)³ = 2
ref02
1214928³+3480205³-3528875³ = 2 is not related to the previous formula

Yasutoshi Kohmoto recently discovered a new formula:

x₀=9*a⁴
y₀=-9*a⁴+3*a*b³
z₀=-9*a³*b+b⁴
u₀=b⁴

x₁=9*a⁴
y₁=-9*a⁴-3*a*b³
z₁=9*a³*b+b⁴
u₁=b⁴

for n=2
x_n=(432*a⁶-2*b⁶)*x_n-1-x_n-2*b⁶-108*a⁴*b⁶
y_n=(432*a⁶-2*b⁶)*y_n-1-y_n-2*b⁶-108*a⁴*b⁶
z_n=(432*a⁶-2*b⁶)*z_n-1-z_n-2*b⁶+216*a⁶*b⁴+4*b¹⁰
u_n=b^6*n-2

for 3<=n
x_n=(432*a⁶-2*b⁶)*x_n-1-x_n-2*b¹²-108*a⁴*b^6*n-6
y_n=(432*a⁶-2*b⁶)*y_n-1-y_n-2*b¹²-108*a⁴*b^6*n-6
z_n=(432*a⁶-2*b⁶)*z_n-1-z_n-2*b¹²+(216*a⁶*b⁴+4*b¹⁰)*b^6*n-12
u_n=b^6*n-2

for all n, we have:
x_n³+y_n³+z_n³=u_n³

ex. If n=4, the identity is as follows :
(725594112*a²²-31912704*b⁶*a¹⁶+194400*b¹²*a¹⁰-279*b¹⁸*a⁴)³
+(-725594112*a²²-241864704*b³*a¹⁹-8398080*b⁶*a¹⁶+2799360*b⁹*a¹³+85536*b¹²*a¹⁰-7776*b¹⁵*a⁷-153*b¹⁸*a⁴+3*b²¹*a)³
+(725594112*b*a²¹+120932352*b⁴*a¹⁸-8398080*b⁷*a¹⁵-839808*b¹⁰*a¹²+23328*b¹³*a⁹+1296*b¹⁶*a⁶-9*b¹⁹*a³+b²² )³
=b⁶⁶

Linear Forms

S_r=0^k-1(-1)^k-1-r*(_r^k-1)*(x+r)^k=k!*x+d where d is an integer relatively prime to x
ref01 p. 325

Other equal sums of like power

3rd degree

(3*a²+5*a*b-5*b²)³
+(4*a²-4*a*b+6*b²)³
+(5*a²-5*a*b-3*b²)³ =(6*a²-4*a*b+4*b²)³ref01 p. 201

(1-(a-3*b)*(a²+3*b²))³+((a+3*b)*(a²+3*b²)-1)³= (a+3*b-(a²+3*b²)²)³+((a²+3*b²)²-a+3*b)³
ref01 p. 200

If a³+b³=c³+d³ then
(a*(a³+2*b³)/(a³-b³))³+(d*(2*c³+d³)/(c³-d³))³
=(b*(2*a³+b³)/(a³-b³))³ +(c*(c³+2*d³)/(c³-d³))³
ref01 p. 333

4th degree

(a²-b²)⁴ +(a² + 2*a*b)⁴
+(2*a*b + b²)⁴ =2*(a² + a*b + b²)⁴
ref01 p. 333

(4*a⁴-b⁴)⁴ +2*(4*a³*b)⁴
+2*(2*a*b³)⁴ =(4*a⁴ + b⁴)⁴
ref01 p. 333

(a⁷+a⁵*b²-2*a³*b⁴+3*a²*b⁵+a*b⁶)⁴
+ (a⁶*b-3*a⁵*b²-2*a⁴*b³+a²*b⁵+b⁷)⁴
= (a⁷+a⁵*b²-2*a³*b⁴-3*a²*b⁵+a*b⁶)⁴
+ (a⁶*b+3*a⁵*b²-2*a⁴*b³+a²*b⁵+b⁷)⁴
ref01 p. 201

f1(a,b)=a
f1(a,b)⁴+f1(b,-a)⁴=f1(a,-b)⁴+f1(b,a)⁴

f2(a,b)=-a¹³+a¹²*b+a¹¹*b²+5*a¹⁰*b³+6*a⁹*b⁴-12*a⁸*b⁵-4*a⁷*b⁶+7*a⁶*b⁷-3*a⁵*b⁸-3*a⁴*b⁹+4*a³*b¹⁰+2*a²*b¹¹-a*b¹²+b¹³
f2(a,b)⁴+f2(b,-a)⁴=f2(a,-b)⁴+f2(b,a)⁴

f3(a,b)=a¹⁹+6*a¹⁷*b²-18*a¹⁵*b⁴+6*a¹⁴*b⁵-5*a¹³*b⁶+12*a¹²*b⁷-12*a¹¹*b⁸+36*a¹⁰*b⁹-24*a⁹*b¹⁰-12*a⁸*b¹¹+19*a⁷*b¹²+36*a⁶*b¹³+6*a⁵*b¹⁴+12*a⁴*b¹⁵-6*a³*b¹⁶+6*a²*b¹⁷+a*b¹⁸
f3(a,b)⁴+f3(b,-a)⁴=f3(a,-b)⁴+f3(b,a)⁴

f4(a,b)=a³¹-a³⁰*b+11*a²⁹*b²+a²⁸*b³+42*a²⁷*b⁴+24*a²⁶*b⁵-19*a²⁵*b⁶-32*a²⁴*b⁷-154*a²³*b⁸-254*a²²*b⁹+266*a²¹*b¹⁰+718*a²⁰*b¹¹+126*a¹⁹*b¹²-303*a¹⁸*b¹³-478*a¹⁷*b¹⁴-830*a¹⁶*b¹⁵+770*a¹⁵*b¹⁶+916*a¹⁴*b¹⁷-738*a¹³*b¹⁸+21*a¹²*b¹⁹+350*a¹¹*b²⁰-434*a¹⁰*b²¹+50*a⁹*b²²+142*a⁸*b²³-91*a⁷*b²⁴+76*a⁶*b²⁵+15*a⁵*b²⁶-3*a⁴*b²⁷+8*a³*b²⁸-8*a²*b²⁹+a*b³⁰-b³¹
f4(a,b)⁴+f4(b,-a)⁴=f4(a,-b)⁴+f4(b,a)⁴
ref04

(z¹³+27*z¹²-214*z¹¹-186*z¹⁰-2481*z⁹+861*z⁸-2804*z⁷-972*z⁶-2481*z⁵-27*z⁴-214*z³+294*z²+z+3)⁴+(3*z¹³-z¹²+294*z¹¹+214*z¹⁰-27*z⁹+2481*z⁸-972*z⁷+2804*z⁶+861*z⁵+2481*z⁴-186*z³+214*z²+27*z-1)⁴
=(-z¹³+27*z¹²+214*z¹¹-186*z¹⁰+2481*z⁹+861*z⁸+2804*z⁷-972*z⁶+2481*z⁵-27*z⁴+214*z³+294*z²-z+3)⁴
+(3*z¹³+z¹²+294*z¹¹-214*z¹⁰-27*z⁹-2481*z⁸-972*z⁷-2804*z⁶+861*z⁵-2481*z⁴-186*z³-214*z²+27*z+1)⁴
ref05

(m¹⁹-m¹⁸-3*m¹⁷-3*m¹⁶+21*m¹⁵-6*m¹⁴-44*m¹³+62*m¹²+15*m¹¹-129*m¹⁰+165*m⁹-129*m⁸ +88*m⁷-46*m⁶+18*m⁵-6*m⁴+12*m³-3*m²+m-1)⁴
+(-m¹⁸+3*m¹⁷-3*m¹⁶-21*m¹⁵+60*m¹⁴-27*m¹³-58*m¹²+75*m¹¹-57*m¹⁰+63*m⁹-63*m⁸+87*m⁷-100*m⁶+66*m⁵-36*m⁴+18*m³-9*m²-1)⁴
=(m¹⁹-m¹⁸-3*m¹⁷-3*m¹⁶+21*m¹⁵-12*m¹⁴-44*m¹³+86*m¹²-93*m¹¹+87*m¹⁰+3*m⁹-135*m⁸+142*m⁷-100*m⁶+72*m⁵-36*m⁴+12*m³-9*m²+m-1)⁴
+(m¹⁸+3*m¹⁷-15*m¹⁶+15*m¹⁵+6*m¹⁴-45*m¹³+82*m¹²-15*m¹¹-123*m¹⁰+171*m⁹-159*m⁸+159*m⁷-98*m⁶+30*m⁵-12*m⁴+3*m²+1)⁴
ref05

5th degree

(-a⁵+75*b⁵)⁵+(a⁵+25*b⁵)⁵
+(a⁵-25*b⁵)⁵
+(10*a³*b²)⁵ +(50*a*b⁴)⁵ =(a⁵+75*b⁵)⁵
ref01 p. 333

(-a²+4*a*b+9*b²)⁵+(a²+8*a*b+3*b²)⁵+2*(3*a²+12*a*b+21*b²)⁵=(a²+12*a*b+23*b²)⁵+(3*a²+16*a*b+17*b²)⁵+(-a²+13*b²)⁵+(3*a²+8*a*b+b²)⁵
ref05

(2*a⁸*b+10*a⁷*b²-20*a⁶*b³+20*a⁵*b⁴-34*a⁴*b⁵-10*a³*b⁶+270*a²*b⁷-20*a*b⁸+682*b⁹)⁵
+(-2*a⁸*b+10*a⁷*b²+20*a⁶*b³+20*a⁵*b⁴+34*a⁴*b⁵-10*a³*b⁶-270*a²*b⁷-20*a*b⁸-682*b⁹)⁵
+(a⁹-22*a⁵*b⁴-125*a³*b⁶-79*a*b⁸)⁵
=(a⁸*b+10*a⁷*b²-10*a⁶*b³+20*a⁵*b⁴-92*a⁴*b⁵-160*a³*b⁶-15*a²*b⁷-320*a*b⁸+341*b⁹)⁵
+(-a⁸*b+10*a⁷*b²+10*a⁶*b³+20*a⁵*b⁴+92*a⁴*b⁵-160*a³*b⁶+15*a²*b⁷-320*a*b⁸-341*b⁹)⁵
+(a⁹-22*a⁵*b⁴+175*a³*b⁶+521*a*b⁸)⁵

(c²*(m⁷+m⁵-2*m³+m)+3*c²*m²-3*a*c*m²+a²*m)⁵
+(c²*(m⁷+m⁵-2*m³+m)-3*c²*m²+3*a*c*m²+a²*m)⁵
+(c²*(m⁶-2*m⁴+m²+1)+3*c²*m⁵+3*a*c*m²+a²)⁵
+(c²*(m⁶-2*m⁴+m²+1)-3*c²*m⁵-3*a*c*m²+a²)⁵
=(c²*(m⁷+m⁵-2*m³+m)+3*c²*m²+3*a*c*m²+a²*m)⁵
+(c²*(m⁷+m⁵-2*m³+m)-3*c²*m²-3*a*c*m²+a²*m)⁵
+(c²*(m⁶-2*m⁴+m²+1)+3*c²*m⁵-3*a*c*m²+a²)⁵
+(c²*(m⁶-2*m⁴+m²+1)-3*c²*m⁵+3*a*c*m²+a²)⁵

(-((2*c⁷+(-b-a)*c⁶+(20*b²+6*a²)*c⁵ +(17*b³+17*a*b²-3*a²*b-3*a³)*c⁴+(2*b⁴+24*a²*b²+6*a⁴) *c³+(17*b⁵+17*a*b⁴+6*a²*b³+6*a³*b²-3*a⁴*b-3*a⁵)*c² +(8*b⁶+18*a²*b⁴+12*a⁴*b²+2*a⁶)*c-b⁷-a*b⁶-3*a²*b⁵-3 *a³*b⁴-3*a⁴*b³-3*a⁵*b²-a⁶*b-a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3 *a²*b⁴+3*a⁴*b²+a⁶)))⁵
+(-((2*c⁷+(b+a)*c⁶+(20*b²+6*a²)*c⁵ +(-17*b³-17*a*b²+3*a²*b+3*a³)*c⁴ +(2*b⁴+24*a²*b²+6*a⁴)*c³ +(-17*b⁵-17*a*b⁴-6*a²*b³-6*a³*b²+3*a⁴*b+3*a⁵)*c²+(8*b⁶+18*a²*b⁴+12*a⁴*b²+2*a⁶)*c+b⁷+a*b⁶+3*a²*b⁵+3 *a³*b⁴+3*a⁴*b³+3*a⁵*b²+a⁶*b+a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3 *a²*b⁴+3*a⁴*b²+a⁶)))⁵
+(((c⁷+(-8*b-a)*c⁶+(3*a²-17*b²)*c⁵ +(-2*b³+17*a*b²-18*a²*b-3*a³)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c³ +(-20*b⁵+17*a*b⁴-24*a²*b³+6*a³*b²-12*a⁴*b-3*a⁵)*c²+(b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)*c-2*b⁷-a*b⁶-6*a²*b⁵-3*a³*b⁴ -6*a⁴*b³-3*a⁵*b²-2*a⁶*b-a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)))⁵
+(-((c⁷+(8*b-a)*c⁶+(3*a²-17*b²)*c⁵ +(2*b³+17*a*b²+18*a²*b-3*a³)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c³ +(20*b⁵+17*a*b⁴+24*a²*b³+6*a³*b²+12*a⁴*b-3*a⁵)*c²+(b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)*c+2*b⁷-a*b⁶+6*a²*b⁵-3*a³*b⁴ +6*a⁴*b³-3*a⁵*b²+2*a⁶*b-a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)))⁵
=(-((2*c⁷+(a-b)*c⁶+(20*b²+6*a²)*c⁵ +(17*b³-17*a*b²-3*a²*b+3*a³)*c⁴ +(2*b⁴+24*a²*b²+6*a⁴)*c³ +(17*b⁵-17*a*b⁴+6*a²*b³-6*a³*b²-3*a⁴*b+3*a⁵)*c²+(8*b⁶+18*a²*b⁴+12*a⁴*b²+2*a⁶)*c-b⁷+a*b⁶-3*a²*b⁵+3*a³*b⁴-3*a⁴*b³+3*a⁵*b²-a⁶*b+a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3 *a²*b⁴+3*a⁴*b²+a⁶)))⁵
+(-((2*c⁷+(b-a)*c⁶+(20*b²+6*a²)*c⁵ +(-17*b³+17*a*b²+3*a²*b-3*a³)*c⁴ +(2*b⁴+24*a²*b²+6*a⁴)*c³ +(-17*b⁵+17*a*b⁴-6*a²*b³+6*a³*b²+3*a⁴*b-3*a⁵)*c²+(8*b⁶+18*a²*b⁴+12*a⁴*b²+2*a⁶)*c+b⁷-a*b⁶+3*a²*b⁵-3 *a³*b⁴+3*a⁴*b³-3*a⁵*b²+a⁶*b-a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)))⁵
+(((c⁷+(a-8*b)*c⁶+(3*a²-17*b²)*c⁵ +(-2*b³-17*a*b²-18*a²*b+3*a³)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c³ +(-20*b⁵-17*a*b⁴-24*a²*b³-6*a³*b²-12*a⁴*b+3*a⁵)*c²+(b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)*c-2*b⁷+a*b⁶-6*a²*b⁵+3*a³*b⁴-6*a⁴*b³+3*a⁵*b²-2*a⁶*b+a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3 *a²*b⁴+3*a⁴*b²+a⁶))⁵
+(-((c⁷+(8*b+a)*c⁶+(3*a²-17*b²)*c⁵ +(2*b³-17*a*b²+18*a²*b+3*a³)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c³ +(20*b⁵-17*a*b⁴+24*a²*b³-6*a³*b²+12*a⁴*b+3*a⁵)*c²+(b⁶+3*a²*b⁴+3*a⁴*b²+a⁶)*c+2*b⁷+a*b⁶+6*a²*b⁵+3*a³*b⁴+6*a⁴*b³+3*a⁵*b²+2*a⁶*b+a⁷)/(c⁶+(3*a²-17*b²)*c⁴ +(-17*b⁴-6*a²*b²+3*a⁴)*c²+b⁶+3 *a²*b⁴+3*a⁴*b²+a⁶)))⁵
ref05

6th degree

(3*a⁴+9*a³*b+18*a²*b²+21*a*b³+9*b⁴)⁶
+(2*a⁴+4*a³*b-5*a²*b²-12*a*b³-9*b⁴)⁶+(-a⁴-10*a³*b-17*a²*b²-12*a*b³)⁶=(a⁴-3*a³*b-14*a²*b²-15*a*b³-9*b⁴)^{6

+}(3*a⁴+8*a³*b+9*a²*b²)⁶+(2*a⁴+12*a³*b+19*a²*b²+18*a*b³+9*b⁴)⁶

(3*a⁵+8*a⁴*b+9*a³*b²-4*a²*b³-9*a*b⁴-2*b⁵)⁶+(-2*a⁵-a⁴*b+12*a³*b²+13*a²*b³+4*a*b⁴-b⁵)⁶+(-a⁵-9*a⁴*b-13*a³*b²-7*a²*b³-7*a*b⁴-3*b⁵)⁶
=(2*a⁵+9*a⁴*b+4*a³*b²-9*a²*b³-8*a*b⁴-3*b⁵)⁶+(-3*a⁵-7*a⁴*b-7*a³*b²-13*a²*b³-9*a*b⁴-b⁵)⁶
+(-a⁵+4*a⁴*b+13*a³*b²+12*a²*b³-a*b⁴-2*b⁵)⁶

(16*a⁵+64*a⁴*b+104*a³*b²+116*a²*b³+78*a*b⁴+27*b⁵)⁶
+(16*a⁵+24*a⁴*b+48*a³*b²+52*a²*b³+57*a*b⁴+18*b⁵)⁶
+(16*a⁵+32*a⁴*b+52*a³*b²+48*a²*b³+21*a*b⁴-9*b⁵)⁶
=(16*a⁵+48*a⁴*b+84*a³*b²+76*a²*b³+33*a*b⁴+18*b⁵)⁶+(16*a⁵+56*a⁴*b+112*a³*b²+108*a²*b³+81*a*b⁴+27*b⁵)⁶+(16*a⁵+16*a⁴*b+8*a³*b²-28*a²*b³-18*a*b⁴-9*b⁵)⁶

(2*a⁷+7*a⁶*b+15*a⁵*b²+18*a⁴*b³+9*a³*b⁴+2*a²*b⁵+a*b⁶+b⁷)⁶
+(a⁷+a⁶*b-4*a⁵*b²-17*a⁴*b³-22*a³*b⁴-15*a²*b⁵-7*a*b⁶-2*b⁷)⁶+(-a⁷-6*a⁶*b-17*a⁵*b²-21*a⁴*b³-17*a³*b⁴-11*a²*b⁵-6*a*b⁶-b⁷)⁶
=(-a⁷-a⁶*b-2*a⁵*b²-9*a⁴*b³-18*a³*b⁴-15*a²*b⁵-7*a*b⁶-2*b⁷)⁶
+(a⁷+6*a⁶*b+11*a⁵*b²+17*a⁴*b³+21*a³*b⁴+17*a²*b⁵+6*a*b⁶+b⁷)⁶
+(2*a⁷+7*a⁶*b+15*a⁵*b²+22*a⁴*b³+17*a³*b⁴+4*a²*b⁵-a*b⁶-b⁷)⁶

(15*a⁸-96*a⁷*b+295*a⁶*b²-628*a⁵*b³+920*a⁴*b⁴-880*a³*b⁵+608*a²*b⁶-320*a*b⁷+64*b⁸)⁶
+(-25*a⁸+163*a⁷*b-483*a⁶*b²+896*a⁵*b³-1220*a⁴*b⁴+1200*a³*b⁵-784*a²*b⁶+384*a*b⁷-128*b⁸)⁶
+(-5*a⁸+22*a⁷*b-84*a⁶*b²+284*a⁵*b³-620*a⁴*b⁴+880*a³*b⁵-816*a²*b⁶+512*a*b⁷-192*b⁸)⁶
=(-25*a⁸+162*a⁷*b-480*a⁶*b²+908*a⁵*b³-1220*a⁴*b⁴+1200*a³*b⁵-816*a²*b⁶+320*a*b⁷-64*b⁸)⁶
+(15*a⁸-109*a⁷*b+309*a⁶*b²-544*a⁵*b³+800*a⁴*b⁴-960*a³*b⁵+832*a²*b⁶-512*a*b⁷+192*b⁸)⁶+(-5*a⁸-12*a⁷*b+143*a⁶*b²-376*a⁵*b³+580*a⁴*b⁴-720*a³*b⁵+656*a²*b⁶-384*a*b⁷+128*b⁸)⁶

(16*a⁹+64*a⁸*b+160*a⁷*b²+308*a⁶*b³+416*a⁵*b⁴+397*a⁴*b⁵+262*a³*b⁶+139*a²*b⁷ +54*a*b⁸+9*b⁹)⁶
+(-16*a⁹-56*a⁸*b-80*a⁷*b²-12*a⁶*b³+119*a⁵*b⁴+234*a⁴*b⁵+233*a³*b⁶+126*a²*b⁷+27*a*b⁸)⁶
+(16*a⁹+96*a⁸*b+236*a⁷*b²+368*a⁶*b³+387*a⁵*b⁴+261*a⁴*b⁵+75*a³*b⁶-41*a²*b⁷-39*a*b⁸-9*b⁹)⁶
=(-16*a⁹-48*a⁸*b-44*a⁷*b²+60*a⁶*b³+225*a⁵*b⁴+290*a⁴*b⁵+231*a³*b⁶+106*a²*b⁷+21*a*b⁸)⁶
+(16*a⁹+88*a⁸*b+208*a⁷*b²+324*a⁶*b³+369*a⁵*b⁴+355*a⁴*b⁵+281*a³*b⁶+149*a²*b⁷+51*a*b⁸+9*b⁹)⁶
+(-16*a⁹-80*a⁸*b-224*a⁷*b²-364*a⁶*b³-360*a⁵*b⁴-199*a⁴*b⁵-34*a³*b⁶+51*a²*b⁷+42*a*b⁸+9*b⁹)⁶

(8*a¹¹+43*a¹⁰*b+167*a⁹*b²+547*a⁸*b³+1296*a⁷*b⁴+1863*a⁶*b⁵+1631*a⁵*b⁶+927*a⁴*b⁷+352*a³*b⁸+63*a²*b⁹-17*a*b¹⁰-5*b¹¹)⁶
+(-a¹¹+39*a¹⁰*b+213*a⁹*b²+610*a⁸*b³+1005*a⁷*b⁴+939*a⁶*b⁵+587*a⁵*b⁶+412*a⁴*b⁷+341*a³*b⁸+177*a²*b⁹+45*a*b¹⁰+8*b¹¹)⁶
+(5*a¹¹+38*a¹⁰*b+42*a⁹*b²-155*a⁸*b³-769*a⁷*b⁴-1554*a⁶*b⁵-1744*a⁵*b⁶-1217*a⁴*b⁷-613*a³*b⁸-232*a²*b⁹-50*a*b¹⁰-b¹¹)⁶
=(-5*a¹¹-17*a¹⁰*b+63*a⁹*b²+352*a⁸*b³+927*a⁷*b⁴+1631*a⁶*b⁵+1863*a⁵*b⁶+1296*a⁴*b⁷+547*a³*b⁸+167*a²*b⁹+43*a*b¹⁰+8*b¹¹)⁶+(a¹¹+50*a¹⁰*b+232*a⁹*b²+613*a⁸*b³+1217*a⁷*b⁴+1744*a⁶*b⁵+1554*a⁵*b⁶+769*a⁴*b⁷+155*a³*b⁸-42*a²*b⁹-38*a*b¹⁰-5*b¹¹)⁶
+(-8*a¹¹-45*a¹⁰*b-177*a⁹*b²-341*a⁸*b³-412*a⁷*b⁴-587*a⁶*b⁵-939*a⁵*b⁶-1005*a⁴*b⁷-610*a³*b⁸-213*a²*b⁹-39*a*b¹⁰+b¹¹)⁶ref06

Miscellaneous

Liouville's identity (a-b)⁴+(a+b)⁴+(a-c)⁴+(a+c)⁴+(a-d)⁴+(a+d)⁴+(b-c)⁴+(b+c)⁴+(b-d)⁴+(b+d)⁴+(c-d)⁴+(c+d)⁴ =6*(a²+b²+c²+d²)²
ref01 p. 317
Sebastián Martín Ruiz found the following relation: Sum[(-1)^i*Binomial(n,i)*(x-i)^n,{i,0,n}]=n!

References

ref01:An introduction to the theory of numbers 3rd edition G.H. Hardy & E.M. Wright
ref02:Solutions of the diophantine equation x³+y³+z³=k J.C.P. Miller & M.F.C. Woollett Journal of the London mathematical society 1955 volume 30 pp. 101-110
ref03:On the diophantine equation x³+y³+z³=1 D.H. Lehmer Mathematics of computation p. 275-280
ref04:Some new results on equal sums of like powers Simcha Brudno Mathematics of computation volume 25 1969 pp. 877-880
ref05:Geometric aspects of diophantine equations involving equal sums of like powers L.J. Lander American mathematical monthly volume 75 no 6-10 1968 pp. 1061-1073
ref06:On the diophantine equation x₁⁶+x₂⁶+x₃⁶=y₁⁶+y₂⁶+y₃⁶ Jean-Noël Delorme Mathematics of computation volume 59 number 200 october 1992 pages 703-715

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