T(x) = x/d if x ≡ 0 (mod d), otherwise
T(x) = (mx - r) ⁄ d if mx ≡ r (mod d), r ∈ R_{d}.
T(x) = int(m_{i}x ⁄ d) + x_{i}, if x ≡ i (mod d).
Equivalently, if r_{0},...,r_{d-1} satisfy r_{i} ≡ im_{i} (mod d) for i = 0,...,d-1, thenT(x) = (m_{i}x - r_{i}) ⁄ d if x ≡ i (mod d).
The 3x+1 (Collatz 1937) mapping T(x)=x ⁄ 2 if x is even, (3x+1) ⁄ 2 if x is odd, corresponds to m_{0}=1, m_{1}=3, x_{0}=0, x_{1}=1.
Starting in 1982, Tony Watts and I investigated the frequencies of occupation of congruence classes mod m of trajectories arising from generalised 3x+1 functions. (See earlier 3x+1 interests)
Experimentally we observed in cases (a) and (b) below, that (apparently) divergent (ie. unbounded) trajectories were attracted to ergodic sets mod m of a Markov matrix Q_{T}(m) with limiting frequencies that we were later able to predict.
(a) gcd(m_{i}, d) = 1 for 0 ≤ i ≤ d-1, or
(b) gcd(m_{i}, d^{2}) = gcd(m_{i}, d) for 0 ≤ i ≤ d-1 and d divides m.
Here Q_{T}_{i}_{j}(m) = p_{i}_{j}(m) ⁄ d, where p_{i}_{j}(m) is the number of congruence classes mod md which comprise B(i, m) ∩ T^{-1}(B(j, m)), where B(i, m) denotes the congruence class of integers x ≡ i (mod m).(See a program for constructing Q_{T}(m) and finding the ergodic sets and transient classes, using the Fox_Landi algorithm with labels 0,1,...,m-1.)
In the case when neither condition (a) nor (b) holds, it is still possible to get conjectural information using Markov matrices and this was pointed out by George Leigh in 1983.
The asymptotic behaviour of T^{-K}(B(j, m))
Theorem 0. T^{-1}(B(j, m)) is a disjoint union (possibly empty) of ∑^{'} gcd(m_{i}, m) congruence classes (mod md), where M_{i} = (m_{i}i - r_{i} ) ⁄ d and the dash denotes a summation over 0 ≤ i ≤ d-1 and where gcd(m_{i}, m) divides j - M_{i}.x ≡ i (mod d) and (m_{i}x - r_{i} ) ⁄ d ≡ j (mod m)
for i = 0,...,d-1.(m_{i}(i + dy) - r_{i} ) ⁄ d ≡ j (mod m), or m_{i}y ≡ j - M_{i} (mod m).
This congruence is soluble if and only if gcd(m_{i}, m) divides j - M_{i}, in which case there are gcd(m_{i}, m) solutions for y (mod m) and hence for x (mod md).
x ≡ i (mod d) and x ≡ k (mod m)
are soluble if and only if i ≡ k (mod D) and the solution is unique (mod L).
Lemma 2. Let p_{kj}(n, m) be the number of congruence classes mod nd contained in B(k, m) ∩ T^{-1}(B(j, n)), where m divides n.
Then p_{kj}(n, m) is given by the following formula:
Let D = gcd(m, d), L = lcm(d, m) and let S_{k,D} consist of the integers i≡ k (mod D), 0 ≤ i ≤ d-1. Also for i ∈ S_{k,D}, let x_{i} denote the solution of
x ≡ i (mod d) and x ≡ k (mod m), 0 ≤ x ≤ L-1.
Finally let M_{i} = (m_{i} x_{i} - r_{i} ) ⁄ d if i ∈ S_{k,D}. Thenp_{kj}(n, m)=∑^{'} gcd(m_{i}, n, nd ⁄ m),
where the dash denotes summation over those i ∈ S_{k,D} such that (m ⁄ D)gcd(m_{i}, n, nd ⁄ m) divides j - M_{i}.Proof. We have
B(k, m) ∩ T^{-1}(B(j, n)) = {x ∈ ℤ | x ≡ k (mod m) & T(x) ≡ j (mod n)}.
Hence for each i = 0,..., d-1, we have to solve the congruencesx ≡ i (mod d), x ≡ k (mod m), (m_{i}x - r_{i} ) ⁄ d ≡ j (mod n)
Write x = x_{i} + Ly, i ∈ S_{k,D}. Then we have to solve(m_{i}(x_{i} + Ly) - r_{i} ) ⁄ d ≡ j (mod n), or (m_{i}m ⁄ D)y ≡ j - M_{i} (mod n).
This congruence is soluble if and only if t = gcd(m_{i}m ⁄ D, n) divides j - M_{i}, in which case the solution for y is unique (mod n ⁄ t). Nowgcd(m_{i}m ⁄ D, n) = (m ⁄ D)gcd(m_{i}, n, nd ⁄ m)
Hence, in the case of solubility, the solution for x is unique modLn ⁄ t = LDn ⁄ mgcd(m_{i}, n, nd ⁄ m) = nd ⁄ gcd(m_{i}, n, nd ⁄ m).
Hence there are gcd(m_{i}, n, nd ⁄ m) solutions for x (mod nd).Lemma 3. Under conditions (a) or (b) above, if α ≥ 1,
p_{kj}(md^{α}, m) = p_{kj}(m, m),
Proof. By Lemma 2,p_{kj}(md^{α}, m) = ∑^{'} gcd(m_{i}, md^{α}, d^{α+1}),
where the summation is over those i ∈ S_{k,D} satisfying(m ⁄ D) gcd(m _{i}, md^{α}, d^{α+1}) divides j - M_{i}.
Howevergcd(m _{i}, md^{α}, d^{α+1}) = gcd(m_{i}, d^{α} gcd(m, d)).
So if condition (a) holds, the rhs = 1; while if condition (b) holds, the rhs is equal to gcd(m_{i}, d^{α+1}) = gcd(m_{i}, d).Theorem 1. Write p_{kj}(m, m)=p_{kj}(m). Then under conditions (a) or (b) above, the cylinder
B(i_{0}, m) ∩ T^{-1}(B(i_{1}, m)) ∩ ··· ∩ T^{-K}(B(i_{K}, m))
consists of p_{i0i1}(m) ··· p_{iK-1iK}(m) congruence classes (mod md^{K}).Proof. We argue by induction.
B(i_{0}, m) ∩ T^{-1}(B(i_{1}, m)) ∩ ··· ∩ T^{-(K+1)}(B(i_{K+1}, m)) = B(i_{0}, m) ∩ T^{-1}{B(i_{1}, m) ∩ ··· ∩ T^{-K}(B(i_{K+1}, m))}
Now by the induction hypothesis, B(i_{1}, m) ∩ ··· ∩ T^{-K}(B(i_{K+1}, m)) consists p_{i1i2}(m) ··· p_{iKiK+1}(m) congruence classes (mod md^{K}).p_{i0L}(md^{K}, m) = p_{i0L}(m, m) = p_{i0i1}(m, m).
Hence B(i_{0}, m) ∩ T^{-1}{B(i_{1}, m) ∩ ··· ∩ T^{-K}(B(i_{K+1}, m))} is a disjoint union p_{i0i1}(m) p_{i1i2}(m) ··· p_{iKiK+1}(m) congruence classes (mod md^{K+1}) and the induction goes through.Corollary 1. Under conditions (a) or (b), if p_{Kjk}(m) is the number of congruence classes (mod md^{K}) contained in B(j, m) ∩ T^{-K}(B(k, m)), then
[p_{Kjk}(m)]=[p_{jk}(m)]^{K}.
Theorem 2. Let q_{jk}(m) = p_{jk}(m) ⁄ d. Then Q_{T}(m) = [q_{jk}(m)] is a Markov matrix, i.e.. a matrix with non-negative entries, each of whose row sums equals 1.Proof.
B(j, m) = B(j, m) ∩ ℤ= B(j, m) ∩ T^{-1}(ℤ) = ∪_{0≤k≤m-1}{B(j, m) ∩ T^{-1}(B(k, m))},
Hence B(j, m) is a disjoint union of ∑_{0≤k≤m-1}p_{jk}(m) congruence classes (mod md).q_{ij}(m) = gcd(m_{i}, d) ⁄ d, if j ≡ T(i) (mod (m ⁄ d)gcd(m_{i}, d)), otherwise q_{ij}(m) = 0.
Hence in the relatively prime case, if d divides m, the formula for q_{ij}(m) reduces toq_{ij}(m) = 1 ⁄ d, if j ≡ T(i) (mod m ⁄ d), otherwise 0.
Example. (3x+1 mapping, m = 3.)
½ | 0 | ½ | |
Q_{T}(3) = | 0 | 0 | 1 |
0 | ½ | ½ |
Under the row/column relabelling (0, 1, 2) → (1, 2, 0),
0 | 1 | 0 | |
Q_{T}(3) → A = | ½ | ½ | 0 |
0 | ½ | ½ |
(1 + (-1)^{n} ⁄ 2^{n-1}) ⁄ 3 | (2 - (-1)^{n} ⁄ 2^{n-1}) ⁄ 3 | 0 | |
A^{n} = | (1 - (-1)^{n} ⁄ 2^{n}) ⁄ 3 | (2 + (-1)^{n} ⁄ 2^{n}) ⁄ 3 | 0 |
(1 + (-1)^{n} ⁄ 2^{n+1}) ⁄ 3 - 1 ⁄ 2^{n+1} | (2 - (-1)^{n} ⁄ 2^{n+1}) ⁄ 3 - 1 ⁄ 2^{n+1} | 1 ⁄ 2^{n} |
Example. (Relatively-prime case)
If A = B(j, d^{a}) and B = B(k, d^{b}), then T^{-k}(A) ∩ B is a disjoint union of d^{k-b} congruence classes (mod d^{k+a}) if k ≥ b.
(This fact is the basis of the strongly-mixing property, when the mapping is extended to the d-adic integers.
Proof. Express A and B as cyclinders.
Irreducible closed sets of Q_{T}(m) and ergodic sets of T
A non-empty subset S of {0,...,m-1} is called a closed set of Q_{T}(m) if j ∈ S and q_{jk} > 0 ⇒ k ∈ S.These are in 1-1 correspondence with the T-invariant subsets of ℤ composed of congruence classes (mod m):
S={j_{1},...,j_{n}} ↔ Φ(S) = B(j_{1}, m) ∪...∪ B(j_{n}, m).
Proof. Suppose S is closed. We show S' = Φ(S) is T-invariant.We call the minimal T-invariant subsets of ℤ consisting of congruence classes (mod m) ergodic sets (mod m) of T. Clearly these are in 1-1 correspondence with the irreducible closed sets of Q_{T}(m).
The set {0,...,m-1} decomposes into a disjoint union of t irreducible sets and other elements called transient.
Correspondingly ℤ decomposes into a union of ergodic sets (mod m) and a collection of transient congruence classes.
With a suitable relabelling of the rows and columns, Q_{T}(m) can be transformed into the following form:
A_{1} | 0 | ··· | ··· | 0 |
0 | A_{2} | ··· | ··· | 0 |
··· | ··· | ··· | ··· | 0 |
0 | 0 | ··· | A_{t} | 0 |
B_{1} | B_{2} | ··· | B_{t} | C |
where matrices A_{1},...,A_{t} and C correspond to the irreducible closed sets and transient classes, respectively.
Theorem 3. If gcd(m_{i}, m) = 1 for i = 0,...,d-1, then Q_{T}(m) is doubly stochastic, i.e., each column sum is equal to 1.
Proof.
T^{-1}(B(k, m)) = ∪_{j=0,...,m-1} B(j, m) ∩ T^{-1}(B(k, m)).
Hence T^{-1}(B(k, m)) is a disjoint union of ∑_{j=0,...,m-1}p_{jk}(m) congruence classes (mod md).∑_{j=0,...,m-1}p_{jk}(m) = d.
Theorem 4. If m divides m_{i}, where gcd(m, d) = 1, then Q_{T}(m) has only one irreducible closed set. Moreover if gcd(m_{i}, d) = 1 for i = 0,...,d-1, then the corresponding submatrix A is primitive, i.e., a suitable power is positive.T(x)= (m_{i}(amd + bd + i) - r_{i}) ⁄ d = amm_{i} + m_{i}b + M_{i} ≡ M_{i} (mod m).
But if 0 ≤ j ≤ m-1, there exists an integer b such thatamd + bd + i ≡ j (mod m).
Hence B(j, m) ∩ T^{-1}(B(M_{i}, m)) ≠ ∅ and so q_{jMi} > 0. Thus all the elements in column M_{i} (mod m) are positive and hence by examining the above normal form of Q_{T}(m), we see that t = 1.For the second part, assume gcd(m_{i}, d) = 1 for i = 0,...,d-1. By standard theory, if A is not primitive, it is periodic with period s and A^{s} has s irreducible closed sets. However it is easy to see from Corollary 1 that
Q_{Ts}(m) = (Q_{T}(m))^{s}.
Hence Q_{Ts}(m) has s irreducible closed sets.However T^{s} has d^{s} branches and associated "m_{i}" equal to m_{i1} ··· m_{is}, where 0 ≤ i_{1},...,i_{s} ≤ d-1; also each of these numbers is relatively prime to d^{s}. Hence by the first part of the proof, Q_{Ts}(m) has but one irreducible closed set.
Corollary 2. If m divides m_{i}, where gcd(m_{j}, d) = 1 for j = 0,...,d-1, then Q(m^{k}) has only one irreducible closed set.
Moreover the corresponding submatrix is primitive.
Proof. Suppose gcd(m_{i}, d) = 1 for i = 0,...,d-1. Then the d^{k} cylinders
B(i_{0}, d) ∩ T^{-1}(B(i_{1}, d)) ∩ ··· ∩ T^{-(k-1)}(B(i_{k-1}, d))
are precisely the d^{k} congruence classes (mod d^{k}).Now suppose m divides m_{i}. Then
B(i, d) ∩ T^{-1}(B(i, d)) ∩ ··· ∩ T^{-(k-1)}(B(i, d)) = B(j, d^{k})
for some j and henceT^{h}(j) ≡ i (mod d) for h = 0,...,k-1.
Consequently m_{i}^{k} is one of the "m_{i}" for T^{k}. Also m^{k} divides m_{i}^{k}. So applying the previous theorem, we deduce that Q_{Tk}(m^{k}) has only one irreducible closed set.However from Corollary 1,
Q_{Tk}(m^{k}) = (Q_{T}(m^{k}))^{k},
so it follows that Q_{T}(m^{k}) has precisely one irreducible closed set.Remark. A complete description of the ergodic sets in the relatively-prime case was done in paper [5].
Theorem. Let N_{1} be the set of positive integers composed of primes which divide at least one m_{i} and let N_{2} be the set of positive integers which are relatively prime to each m_{i}. Also for 0 ≤ i < j ≤ d-1 let
Δ_{i,j} = r_{j}(d-m_{i}) - r_{i}(d-m_{j})
and Δ = gcd_{0≤i<j ≤d-1} Δ_{i,j}.Behaviour of divergent trajectories
Experimentally, it appears that divergent trajectories occupy congruence classes with limiting frequencies. i.e.,lim_{N → ∞} #{k < N | T^{k}(x) ≡ j (mod m)} / N exists,
if |T^{k}(x)| → ∞.It is not difficult to show that if limiting frequencies f_{0},...,f_{d-1} (mod d) exist, then
|T^{k}(x)|^{1/k} → ( |m_{0}|^{f0}···|m_{d-1}|^{fd-1} ) / d,
as k → ∞.T^{k}(x) = (x ⁄ d^{k}) ∏_{0 ≤ i ≤ k-1}m_{i}(x) ∏_{0 ≤ i ≤ k-1}(1 - r_{i}(x) / {m_{i}(x)T^{i}(x)}).
Here (i) we are assuming that T^{i}(x) ≠ 0 and (ii) if T^{i}(x) ≡ j (mod d), we let m_{i}(x) = m_{j} and r_{i}(x) = r_{j}.)
Conjecture 1.
Under the conditions (a) gcd(m_{i}, d) = 1 for 0 ≤ i ≤ d-1, or (b) gcd(m_{i}, d^{2}) = gcd(m_{i}, d) for 0 ≤ i ≤ d-1 and d divides m, a divergent trajectory {T^{k}(x)} will eventually enter some ergodic set S (mod m).
If B(j, m) ⊆ S, the limiting frequency
lim_{N → ∞}(1 ⁄ N)#{k < N | T^{k}(x) ≡ j (mod m)} = μ_{S}(j, m),
where μ_{S}(j, m) is the j (mod m) component of the stationary vector of Q(m) for the irreducible closed set corresponding to S.A standard result from the theory of Markov matrices tells us that
μ_{S}(j, m) = lim_{N → ∞}(1 ⁄ N) ∑_{k < N} card_{mdk}(T^{-k}(B(j, m)) ∩ S) / card_{mdk}(S),
where t = card_{mdk}(A) means that A is composed of t congruence classes (mod md^{k}).Without the summation sign, this limit is perhaps what one would expect the limiting frequency to be.
Conjecture 2. Let S be an ergodic set (mod d). Then
∏_{B(i, d)⊆ S} |m_{i} ⁄ d|^{μS(i, m)} < 1,
then all trajectories starting in S will eventually become periodic.∏_{B(i, d)⊆ S} |m_{i} ⁄ d|^{μS(i, m)} > 1,
then almost all trajectories starting in S will be divergent. i.e., except for an exceptional set S of integers n satisfying #{n ∈ S | -X ≤ n ≤ X} = o(X).Conjecture 3. Suppose gcd(m_{i}, d) = 1 for 0 ≤ i ≤ d-1. Then
lim_{N → ∞}(1 ⁄ N)#{k < N | T^{k}(x) ≡ j (mod d^{a})} = 1 ⁄ d^{a}.
Equivalently, since B(j_{0}, d) ∩ T^{-1}(B(j_{1}, d)) ∩ ··· ∩ T^{-(a-1)}(B(j_{a-1}, d)) runs over the d^{a} congruence classes (mod d^{a}),lim_{N → ∞}(1 ⁄ N)#{k < N | T^{k}(x) ≡ j_{0} (mod d),...,T^{k+a-1}(x) ≡ j_{a-1} (mod d)} = 1 ⁄ d^{a}.
Remarks
Examples of generalized 3x+1 mappings of relatively prime type
(a) 0, 0;
(b) 1, 2, 1;
(c) -1, -1;
(d) -5, -7, -10, -5;
(e) -17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34, -17.
It can be shown that if 3 does not divide m, then ℤ is the only ergodic set S (mod m), whereas is 3 divides m, ℤ-3ℤ is the only ergodic set S (mod m). (The reader can experiment with a BCMATH program.)
In both cases, the matrix corresponding to S is primitive.
Experimentally, we find that as k varies, the number of cycles is unbounded, as is the greatest length of a cycle.
In fact, in answer to a question from the author, Alan Jones showed that
with k = 2^{N} - 9, N = 2c + 1, the integers 2^{r} + 3, 1 ≤ r ≤ c generate c different cycles each of length N.
The number of cycles for the mappings with k = 5^{t} appears to grow monotonically, as does the maximum cycle length.
T(x) = | (x + a)/2 | if x ≡ 0 (mod 2) |
(3x + b)/2 | if x ≡ 1 (mod 2), |
x ⁄ 3 | if x ≡ 0 (mod 3) | |
T(x) = | (2x - 2) ⁄ 3 | if x ≡ 1 (mod 3) |
(13x - 2) ⁄ 3 | if x ≡ 2 (mod 3). |
x ⁄ 4 | if x ≡ 0 (mod 4) | |
T(x) = | (3x - 3) ⁄ 4 | if x ≡ 1 (mod 4) |
(5x - 2) ⁄ 4 | if x ≡ 2 (mod 4) | |
(17x - 3) ⁄ 4 | if x ≡ 3 (mod 4) |
Divergent trajectories appear to occupy the congruence classes (mod 5) with limiting frequencies 0, 1 ⁄ 15, 2 ⁄ 15, 8 ⁄ 15 and 4 ⁄ 15.
½ | 0 | 0 | ½ | 0 | |
0 | 0 | 0 | 1 | 0 | |
Q(5) = | 0 | ½ | 0 | ½ | 0 |
0 | 0 | 0 | ½ | ½ | |
0 | 0 | ½ | ½ | 0 |
On relabelling the states as 1, 2, 3, 4, 0, Q(5) transforms to canonical form
0 | 0 | 1 | 0 | 0 |
½ | 0 | ½ | 0 | 0 |
0 | 0 | ½ | ½ | 0 |
0 | ½ | ½ | 0 | 0 |
0 | 0 | ½ | 0 | ½ |
The zero limiting frequency is easily explained: if x = 2c + 1, then T(x) = 5c + 3, so clearly every trajectory starting from a non-zero integer will eventually reach B(3, 5).
1 | 0 | 0 | |
Q(3) = | 0 | ½ | ½ |
0 | ½ | ½ |
Examples of generalized 3x+1 mappings satisfying condition (b), but not of relatively prime type
x ⁄ 3 - 1 | if x ≡ 0 (mod 3) | |
T(x) = | (x + 5) ⁄ 3 | if x ≡ 1 (mod 3) |
10x - 5 | if x ≡ 2 (mod 3) |
⅓ | ⅓ | ⅓ | |
Q(3) = | ⅓ | ⅓ | ⅓ |
1 | 0 | 0 |
Also
(1 ⁄ 3)^{½}(1 ⁄ 3)^{¼}(30 ⁄ 3)^{¼} < 1,
so we expect all trajectories to eventually cycle. In fact there appear to be five cycles, starting with values 0, 5, 17, -1, -4.George Leigh (1983) pointed out that cycling can also be predicted by considering the related mapping T', where
T'(x) = T(x) if x ≡ 0 or 1 (mod 3)
T'(x) = T^{2}(x) = (10x - 8) ⁄ 3 if x ≡ 2 (mod 3).
2x ⁄ 3 | if x ≡ 0 (mod 3) | |
T(x) = | (4x - 1) ⁄ 3 | if x ≡ 1 (mod 3) |
(4x + 1) ⁄ 3 | if x ≡ 2 (mod 3) |
T is a 1-1 mapping and its inverse S is the four-branched mapping
3x ⁄ 2 | if x ≡ 0 (mod 4) | |
S(x) = | (3x + 1) ⁄ 4 | if x ≡ 1 (mod 4) |
3x ⁄ 2 | if x ≡ 2 (mod 4) | |
(3x - 1) ⁄ 4 | if x ≡ 3 (mod 4) |
½ | 0 | ½ | 0 | |
Q(4) = | ¼ | ¼ | ¼ | ¼ |
0 | ½ | 0 | ½ | |
¼ | ¼ | ¼ | ¼ |
Also
(6 ⁄ 4)^{¼} (3 ⁄ 4)^{¼} (6 ⁄ 4)^{¼} (3 ⁄ 4)^{¼} > 1,
so again we expect most trajectories of S to be divergent.
3x ⁄ 2 | if x ≡ 0 (mod 4) | |
T(x) = | (x + 1) ⁄ 2 | if x ≡ 1 (mod 4) |
x ⁄ 2 + 1 | if x ≡ 2 (mod 4) | |
(7x + 1) ⁄ 2 | if x ≡ 3 (mod 4) |
½ | 0 | ½ | 0 | |
0 | ½ | 0 | ½ | |
½ | 0 | ½ | 0 | |
0 | ½ | 0 | ½ |
½ | ½ | 0 | 0 |
½ | ½ | 0 | 0 |
0 | 0 | ½ | ½ |
0 | 0 | ½ | ½ |
The weighted product for S_{1} is (6 ⁄ 4)^{½}(2 ⁄ 4)^{½} < 1 and for S_{2} is (2 ⁄ 4)^{½}(14 ⁄ 4)^{½} > 1.
So we expect all trajectories starting with an even integer to eventually cycle, while most trajectories starting with an odd integer should be divergent.
3x ⁄ 2 + 1 | if x ≡ 0 (mod 4) | |
T(x) = | (x - 1) ⁄ 2 | if x ≡ 1 (mod 4) |
x ⁄ 2 | if x ≡ 2 (mod 4) | |
(7x - 1) ⁄ 2 | if x ≡ 3 (mod 4) |
0 | ½ | 0 | ½ | |
Q(4) = | ½ | 0 | ½ | 0 |
0 | ½ | 0 | ½ | |
½ | 0 | ½ | 0 |
The stationary vector is (¼,¼,¼,¼) and the weighted product is (6 ⁄ 4)^{¼}(2 ⁄ 4)^{¼}(2 ⁄ 4)^{¼}(14 ⁄ 4)^{¼} > 1.
Hence we expect almost all trajectories to diverge.
2x | if x ≡ 0 (mod 3) | |
T(x) = | (7x + 2) ⁄ 3 | if x ≡ 1 (mod 3) |
(x - 2) ⁄ 3 | if x ≡ 2 (mod 3) |
1 | 0 | 0 | |
Q(3) = | ⅓ | ⅓ | ⅓ |
⅓ | ⅓ | ⅓ |
Last modified 17th August 2011