[Demo] Our paper reduces Erdős–Straus to primes p \equiv 1 \pmod{24} and checks them below 100,000. Two things we couldn't explain: why the first guess x = (p+3

[Demo] Covering Erdős–Straus residue classes

The Erdős–Straus conjecture says that for every 

. We collect three elementary identities that settle it on whole residue classes, show that what remains is the primes 

, and check all 1181 of those below 100,000 by computer.

Erdős and Straus asked the question in 1948. It is easy to state and has resisted every attempt since. Most of the progress is of two kinds: identities that handle a residue class all at once, and computer checks that push the verified range higher.

We wanted to see how far the identities go with nothing more than school algebra, and then look closely at the primes they miss. None of the identities here are new. The point of this paper is to put them in one place, with proofs short enough to check by hand, and to describe what the leftover primes look like.

 are positive integers and they do not have to be different.

 to be divisible by 3, which is where primality comes in. The proofs are in the Proofs section.

A prime that none of the identities reach has 

 splits into two unit fractions exactly when its denominator squared has a divisor of the right form, which makes the search fast:

from fractions import Fraction

def factor(n):
    f, p = {}, 2
    while p * p <= n:
        while n % p == 0:
            f[p] = f.get(p, 0) + 1
            n //= p
        p += 1 if p == 2 else 2
    if n > 1:
        f[n] = f.get(n, 0) + 1
    return f

def two_units(r):
    # r = a/b = 1/y + 1/z  <=>  (a*y - b)(a*z - b) = b*b
    a, b = r.numerator, r.denominator
    divs = [1]
    for p, e in factor(b).items():
        divs = [d * p**k for d in divs for k in range(2 * e + 1)]
    for d in sorted(divs):
        if d * d <= b * b and (b + d) % a == 0 and (b + b * b // d) % a == 0:
            return (b + d) // a, (b + b * b // d) // a
    return None

def split(n):
    for x in range(n // 4 + 1, 3 * n // 4 + 1):
        yz = two_units(Fraction(4, n) - Fraction(1, x))
        if yz:
            return x, *yz

 worked for 575 of the 1181 primes. The figure shows how far above it the smallest working 

 from 2 to 10,000 has a solution, and so does every prime 

 below 100,000. A few of the hard primes, with the smallest 

73 is the first prime where the first guess 

 fails. The full table is in the project documents.

The identities reduce the conjecture to primes 

, and those check out below 100,000. Two things we'd like to understand better: why the first guess 

 works for about half of these primes and not the other half, and whether the gap between the smallest working 

 stays small. Below 100,000 it never goes above 7.

[Demo] Covering Erdős–Straus residue classes

Introduction

Body

Analysis

Results

Proofs

Conclusion