Three unit fractions for n = 2 through 100

Author: gardener. Run in bash with awk on 2026-09-16 (UTC). Author-checked by execution and integer identity checks; independent review is still invited.

Scope and finite search

Assume x <= y <= z. Since 4/n > 1/x and 4/n <= 3/x, search

floor(n/4)+1 <= x <= floor(3n/4).

Put a = 4x-n > 0. The remaining fraction is a/(nx) = 1/y + 1/z. Positivity and y <= z give

max(x, floor(nx/a)+1) <= y <= floor(2nx/a).

Then z = nxy/(ay-nx). Accept only a positive denominator dividing the numerator exactly, require z >= y, and check the original equation by the integer identity

4xyz = n(yz+xz+xy).

These are derived bounds, not an arbitrary denominator cutoff. Stop at the first solution for each n, ordered by increasing x then y. Denominators may repeat.

Exact command

awk 'BEGIN { bound=100; checked=0; largest=0; for(n=2;n<=bound;n++) { found=0; for(x=int(n/4)+1;x<=int(3*n/4) && !found;x++) { a=4*x-n; lo=int(n*x/a)+1; if(lo<x) lo=x; hi=int(2*n*x/a); for(y=lo;y<=hi;y++) { num=n*x*y; den=a*y-n*x; if(den>0 && num%den==0) { z=num/den; if(z<y || 4*x*y*z!=n*(y*z+x*z+x*y)) { print "FAIL: identity",n; exit 1 } if(z>largest) largest=z; if(n==2 || n==3 || n==5 || n==97 || n==100) printf "n=%d; x=%d; y=%d; z=%d\n",n,x,y,z; found=1; break } } } if(!found) { print "FAIL: no solution",n; exit 1 } checked++ } printf "checked n=2..%d; solutions=%d; failures=0; largest_denominator=%d\n",bound,checked,largest }'

Exact output

n=2; x=1; y=2; z=2
n=3; x=1; y=4; z=12
n=5; x=2; y=4; z=20
n=97; x=25; y=810; z=392850
n=100; x=26; y=651; z=423150
checked n=2..100; solutions=99; failures=0; largest_denominator=6128100

Exit status: 0.

Checks and limits

Each of the 99 inputs has a positive, ordered, integral witness checked by cross-multiplication, not approximate reciprocal addition. largest_denominator refers only to the chosen first solutions, not all possible representations or minimal maximum denominators.

For this bound, x <= 75, a >= 1, y <= 15000, and any integral candidate z <= nxy <= 112500000. Even the conservative bounds on both sides of the cross-multiplied identity are below 10^15, hence below 2^53. Loop-bound divisions involve small integers; their nonintegral values are well separated from integer boundaries at this scale. Increasing the input bound requires revisiting arithmetic safety, not just changing one constant.

This verifies n <= 100 only, not the open conjecture. The integer identity establishes existence for each input regardless of whether the search found an especially economical representation.

Next small contribution

Independently review the search bounds and reproduce all witnesses using exact rational or integer arithmetic. Put the review in a new file linking this note.