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[ May 22, 2024 ] [ Last update December 29, 2024 ]
Manually searching for Sums of Five Fifthpowers ⩽ 1000
By Patrick De Geest

∑ = 

5 + 

5 + 

5 + 

5 + 

5  

+/-   +/-   +/-   +/-   +/-
random   random   random   random   random
Randomize first four terms and adjust last one random scope

If you find a solution that is not present in this list of  Sums of Five Fifthpowers
please mail me the new find and I will add it to the list.

Following is some Pari/GP code that automates the search for these sums of five fifthpowers.
Lowering the random scope value results in finding smaller solutions much sooner. Using
higher random scope values will make a find much rarer but more satisfying.


\\ This is my PARI/GP program code for WONplate 230 by PDG [May 23, 2024] → rnd5pow.gp

f="D:/Teksten2024/Random-so5p.txt";
{
scope=1500;

print("Random between -",scope," and +",scope," limit up to 1000"); print();
write(f, "Random between -",scope," and +",scope," limit up to 1000"); write(f,);

while(1==1,

y=round(random(0.1)*scope*2-scope);
x=round(random(0.1)*scope*2-scope);
w=round(random(0.1)*scope*2-scope);
v=round(random(0.1)*scope*2-scope);

s4p=v^5+w^5+x^5+y^5;

z=round(sqrtn(abs(s4p),5));

z=-1*sign(s4p)*z; n=s4p+z^5;

if(v+w==0||v+x==0||v+y==0||v+z==0||w+x==0||w+y==0||w+z==0||x+y==0||x+z==0||y+z==0, n=99999);

if(abs(n)<=1000,

	a=[v,w,x,y,z];
	if(abs(a[1])>abs(a[2]),sw=a[1];a[1]=a[2];a[2]=sw);
	if(abs(a[2])>abs(a[3]),sw=a[2];a[2]=a[3];a[3]=sw);
	if(abs(a[3])>abs(a[4]),sw=a[3];a[3]=a[4];a[4]=sw);
	if(abs(a[4])>abs(a[5]),sw=a[4];a[4]=a[5];a[5]=sw);
	if(abs(a[1])>abs(a[2]),sw=a[1];a[1]=a[2];a[2]=sw);
	if(abs(a[2])>abs(a[3]),sw=a[2];a[2]=a[3];a[3]=sw);
	if(abs(a[3])>abs(a[4]),sw=a[3];a[3]=a[4];a[4]=sw);
	if(abs(a[1])>abs(a[2]),sw=a[1];a[1]=a[2];a[2]=sw);
	if(abs(a[2])>abs(a[3]),sw=a[2];a[2]=a[3];a[3]=sw);
	if(abs(a[1])>abs(a[2]),sw=a[1];a[1]=a[2];a[2]=sw);

	vv=a[1];ww=a[2];xx=a[3];yy=a[4];z=a[5];

	print1(n," = ");
	if(vv<0, print1("(",vv,")"), print1(vv));
	print1("^5+");
	if(ww<0, print1("(",ww,")"), print1(ww));
	print1("^5+");
	if(xx<0, print1("(",xx,")"), print1(xx));
	print1("^5+");
	if(yy<0, print1("(",yy,")"), print1(yy));
	print1("^5+");
	if(z<0, print1("(",z,")"), print1(z)); 
	print1("^5");
	print;

	write1(f, n," = ");
	if(vv<0, write1(f, "(",vv,")"), write1(f, vv));
	write1(f, "^5+");
	if(ww<0, write1(f, "(",ww,")"), write1(f, ww));
	write1(f, "^5+");
	if(xx<0, write1(f, "(",xx,")"), write1(f, xx));
	write1(f, "^5+");
	if(yy<0, write1(f, "(",yy,")"), write1(f, yy));
	write1(f, "^5+");
	if(z<0, write1(f, "(",z,")"), write1(f, z)); 
	write1(f, "^5");
	write(f,);

));
}


\\ This is my PARI/GP program code for WONplate 230 by PDG [May 23, 2024] → so5p.gp
\\ Instead of random picks this time all combinations from -b up to b are scanned,
\\ for one particular value n.
\\ But this comes with a price namely it runs for a very long time...

f="D:/Teksten2024/Vijfdemachten666.txt";
{
n=666 ;

b=50;

print("\e[38;5;92m",n," z<=",b,"\e[38;5;37m");print;
write(f, n" z<="b);write(f,);

for(y=-b,b,
for(x=-b,b,
for(w=-b,b,
for(v=-b,b,

fif=n-v^5-w^5-x^5-y^5; 

fiff=sqrtn(abs(fif),5);

z=floor(fiff*sign(fif));

if(fiff==floor(fiff),
\\ if(fiff==floor(fiff)&&v+w<>0&&v+x<>0&&v+y<>0&&v+z<>0&&w+x<>0&&w+y<>0&&w+z<>0&&x+y<>0&&x+z<>0&&y+z<>0,

	\\[ COPY THE BLUE SORTING/OUTPUT CODE FROM ABOVE HERE ]

)))));
}

print(Strchr(13));

Guidance to the world of OEIS sequences about at most 5 sums of fifth powers.


(A236067) Fifth powers: a(n) = n^5.

(A020896) Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x,y.
(A004842) Numbers that are the sum of at most 2 positive 5th powers.
(A247099) Numbers which are the sum or difference of two fifth powers. {zero allowed}

(A003348) Numbers that are the sum of 3 positive 5th powers.
(A004843) Numbers that are the sum of at most 3 positive 5th powers.
(A344641) Numbers that are the sum of three positive fifth powers in exactly one way.
(A345010) Numbers that are the sum of three {positive} fifth powers in two or more ways.

(A004844) Numbers that are the sum of at most 4 positive 5th powers.
(A344642) Numbers that are the sum of four {positive} fifth powers in exactly one way.
(A344645) Numbers that are the sum of four {positive} fifth powers in exactly two ways.
(A345337) Numbers that are the sum of four {positive} fifth powers in three or more ways.

(A003350) Numbers that are the sum of 5 positive 5th powers.
(A004845) Numbers that are the sum of at most 5 positive 5th powers.
(A342685) Numbers that are the sum of five {positive} fifth powers in two or more ways.
(A342686) Numbers that are the sum of five {positive} fifth powers in exactly two ways.
(A342687) Numbers that are the sum of five {positive} fifth powers in three or more ways.
(A342688) Numbers that are the sum of five {positive} fifth powers in exactly three ways.
(A344518) Numbers that are the sum of five {positive} fifth powers in four or more ways.
(A344519) Numbers that are the sum of five {positive} fifth powers in exactly four ways.
(A345863) Numbers that are the sum of five {positive} fifth powers in five or more ways.
(A346257) Numbers that are the sum of five {positive} fifth powers in exactly five ways.
(A345864) Numbers that are the sum of five {positive} fifth powers in six or more ways.
(A133541) Sum of fifth powers of five consecutive primes.



Sums of Five Fifthpowers [1..1000] Patrick De Geest
Diophantine Equation--5th Powers from MathWorld
A Table of Fifth Powers equal to Sums of Five Fifth Powers, 2009 by James Waldby
Euler's sum of powers conjecture from Wikipedia
Computing Minimal Equal Sums Of Like Powers
A000230 Prime Curios! Prime Puzzle
Wikipedia 230 Le nombre 230














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E-mail address : pdg@worldofnumbers.com