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Timestamp: 2019-04-23 07:57:36+00:00

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Factoring Lipschitz quaternions. Some prime factorizations of Lipschitz quaternions.
Version of 15 November 2011.
§ 1. A quaternion is an ordered quadruple of numbers, interpreted and manipulated according to specific rules. For most applications throughout mathematics the four components are allowed to be any real numbers, but here we examine quaternions where each component is restricted to being an integer — these being the Lipschitz quaternions.
Not studied here are the more general Hurwitz quaternions, where each component can be either a whole integer, or half of an odd integer; with the restriction that wholes and halves cannot be mixed within the same quaternion. The Hurwitz quaternions have essentially unique prime factorizations, while the Lipschitz quaternions often do not. The main feature of this report is a list exhibiting many sets of contrasting Lipschitz prime factorizations.
Notation varies from one author to the next, but we write the four components of a quaternion as a comma-separated list between two square brackets; for example [ +1, −2, +3, 0 ] and [ 0, 0, 0, 0 ]. The four components of quaternion Q are Qh, Qi, Qj and Qk, yielding the notational identity Q ≡ [ Qh, Qi, Qj, Qk ].
§ 2. As with any ordered n-tuples, two quaternions are equal if and only if their respective components are equal. So if P = Q, then Ph = Qh et cetera.
Addition has the usual properties of associativity and commutativity. The additive identity is Z = [ 0, 0, 0, 0 ].
Multiplication by a scalar is straightforward. The scalar should be a real integer.
Using scalar multiplication, Q = Qh × H + Qi × I + Qj × J + Qk × K.
Division by a scalar is of limited avail for integer quaternions. Q ÷ n will equal [ Qh ÷ n, Qi ÷ n, Qj ÷ n, Qk ÷ n ] only when each of the four component quotients turns out to be an integer. Most of the time this does not happen, leaving scalar division undefined in those cases.
Although associative, multiplication is NOT in general commutative, as illustrated by I × J = +K ≠ J × I = −K. Still, multiplication distributes over addition.
Because it serves as the multiplicative identity, H is identified with the real number unity. More broadly, [ Qh, 0, 0, 0 ] equals the real number Qh. Multiplication is assuredly commutative if either factor happens to be a real number — this is simply the multiplication by a scalar mentioned above.
Quaternions of the form [ Qh, Qi, 0, 0 ] (among many other possibilities) are isomorphic to the complex numbers. By analogy thereto, Qh is termed the real part of Q, while Qi, Qj and Qk are the imaginary parts. A celebrated result is that I2 = J2 = K2 = I × J × K = −1, and the fact that −1 has at least three distinct square roots reveals that quaternions are nontrivial in how they extend the complex numbers.
Conjugation provides a special kind of commutativity: (P × Q)* = Q* × P*. Conveniently, Q* × Q = Q × Q* = (Qh)2 + (Qi)2 + (Qj)2 + (Qk)2, which is a real number. Note the difference between (Qh)2 and (Q2)h.
We define the norm of Q as | Q | = Q* × Q, whose square root is the well-known Euclidean norm. Although the Euclidean norm is extrememly useful in much of mathematics, it is not helpful in the study of integer quaternions because it is usually an irrational number. Importantly, the norm is preserved by multiplication: | P × Q | = | P | × | Q |. Because of this, a quaternion whose norm is prime is prime itself. For instance, [ +3, −2, −2, 0 ] is prime because its norm, 17, is a prime real number.
Let R′ = U × R and S′ = S × V. Then Q′ = R′ × S′ is similar to Q = R × S.
With unit W, let R″ = R × W and S″ = W−1 × S. Then Q = R″ × S″ is similar to Q = R × S.
Along the same lines, employing a unit W, T is half-similar to T × W and W × T. Of course, T × W can be rendered as H × T × W to obtain a formal full similarity.
All these similarities are equivalence relations, and are key in shortening the search for factorizations.
The first of those amounts to (P × Q)* = Q* × P*.
where the divisions are guaranteed to be exact.
§ 6. Division of one quaternion by another is of limited usefulness, particularly in the integer version. The usual implementation of the operation is to multiply the dividend by the inverse of the divisor, so that P ÷ Q might equal P × Q−1 (right division). On the other hand, Q−1 × P (left division), which is usually a different value, is just as valid. Moreover, when Q = R × S, a mixture of left and right divisions is possible: R−1 × P × S−1 or S−1 × P × R−1. All four of these will produce answers with the correct norm.
L × M−1 × N × Q−1 × P = (L × M* × N × Q* × P) ÷ ( | M | × | Q | ).
The gist is to multiply everything that can be multiplied, and then divide; this offers the greatest chance that common factors of numerator and denominator will be in the right place to cancel each other and yield a valid answer. Still, the contained scalar divisions will not usually work, and the consequent rarity of factorizations is what makes the search for them interesting.
Although P × P−1 trivially exists, interposing a unit may ruin it. For instance, if P = [ +2, +3, +5, +7 ] then P × I × P−1 = [ 0, −61, +58, +22 ] ÷ 87 which is not an integer quaternion.
Should a divisor be zero, as in Q ÷ Z, the operation inevitably fails.
§ 7. The shape of a quaternion is an ordered quadruple, formed by taking the absolute values of the components of a quaternion, and arranging them in decreasing order. It is written as a comma-separated list within shallow angle brackets. For instance, the shape of [ +5, −2, +7, −7 ] is 〈 7, 7, 5, 2 〉. Repeated numbers are significant, so a shape is not a set.
Two quaternions of the same shape are comorphic, written P ≈ Q.
Under half-similarity, there are at most 48 classes (table 7.3). Whether by W × T or T × W, the same similarity classes are produced.
Some of these representatives will combine if there are repeated numbers in the shape, or if any of the numbers are zero.
A famous theorem assures that there will be at least one shape for every norm.
Two quaternions can be comorphic without being similar, as [ 3, 5, 7, 9 ] and [ 3, 5, 9, 7 ], as no multiplication by units will exchange exactly two elements. On the other hand, similarity does imply comorphism.
§ 9. In the search for "interesting" prime factorizations, we look at those Lipschitz quaternions whose norm is the product of two (real) prime numbers. An example is [ −2, +1, −1, +3 ], whose norm is 15 = 3 × 5; where to seek factors are therefore the shapes 〈 1, 1, 1, 0 〉 and 〈 2, 1, 0, 0 〉.
To be more thorough, we could divide every quaternion of shape 〈 3, 2, 1, 1 〉 by every quaternion of shape 〈 2, 1, 0, 0 〉 and see what integer quotients emerge; but this would yield huge numbers of almost-the-same results, in the worst case entailing 384 × 384 = 147,456 attempts at division, far too many for practical purposes. Instead, we use similarities and comorphisms to get at the heart of the matter without losing generality.
Specifically, if there is a factorization not using the units, there will surely be a comorphic factorization using them. Thus we need consider, as candidates for Q, only dissimilar members of shape X. This reduces, in the worst case, the number of instances from 384 to 12, which are listed in table 7.2. Even better, the rotative transformations at the end of section three, when applied, mean that row 1 of table 7.2 will suffice without loss of generality. This means that the maximum number of candidates for Q falls to four.
While Y might have as many as 384 members, half-similarity means that we need consider no more than 48 candidates (table 7.3) for S.
In our computer program that performed a search, satisfactorily fast performance was achieved by using 4 values for the dividend Q and 48 for the divisor S. Further simplifications are likely possible, particularly since every example found by the search could be realized with a quotient whose components were all nonnegative.
To reduce the number of characters in the large table, commas between quaternion components have been omitted. Also to save space are used the abbreviations 'N' for norm, and 'S' for strong. Two factorizations are strong if neither factor of one factorization is comorphic with either factor of the other. In the excerpt above, the two factorizations of [ 10, 9, 6, 2 ] illustrate this. The alternative is weak, as with the two factorizations of [ 6, 4, 3, 2 ], where [ 2, 0, 0, −1 ] and [ 2, 0, 1, 0 ] are comorphic. Because [ 3, 2, 0, 0 ] and [ 2, 1, 2, 2 ] are not comorphic, the factorizations still qualify for listing.

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