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*a4
y0=-9*a4+3*a
z0=-9*a3+1

x1=9*a4
y1=-9*a4-3*a
z1=9*a3+1

For n= 2, 3, ...
xn=(432*a6-2) * xn-1 - xn-2 - 108 * a4
yn=(432*a6-2) * yn-1 - yn-2 - 108 * a4
zn=(432*a6-2) * zn-1 - zn-2 + 216 * a6 + 4
xn3 + yn3 + zn3 = 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
xn=(432*a6-2*b6)*xn-1-xn-2*b6-108*a4*b6
yn=(432*a6-2*b6)*yn-1-yn-2*b6-108*a4*b6
zn=(432*a6-2*b6)*zn-1-zn-2*b6+216*a6*b4+4*b10
un=b6*n-2

for 3<=n
xn=(432*a6-2*b6)*xn-1-xn-2*b12-108*a4*b6*n-6
yn=(432*a6-2*b6)*yn-1-yn-2*b12-108*a4*b6*n-6
zn=(432*a6-2*b6)*zn-1-zn-2*b12+(216*a6*b4+4*b10)*b6*n-12
un=b6*n-2

for all n, we have:
xn3+yn3+zn3=un3

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

Sr=0k-1(-1)k-1-r*(rk-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)3
ref01 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)6
ref06

# 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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