Septright integral domain.
Version of Friday 26 October 2018.
Dave Barber's other pages.

Part I.

Section 0. This report is an extension of the same author's quintright discussion.

A quintright is an ordered quadruple of integers whose interpretation is based on dividing a right angle into 5 equal parts. Meanwhile, a septright is an ordered sextuple whose interpretation is based on dividing a right angle into 7 equal parts.

Beyond 5 or 7 parts, largely analogous results can be obtained from dividing the right angle into 11, 13, 17, or 19 equal parts. A divisor that is an even number leads to quite a different structure; a multiple of three introduces a rational number (cos 60°) that causes ambiguities.

Septrights make certain numerical patterns more conspicuous than quintrights can, simply because septrights have more components. Going further, an 11-way right-angle division would entail ordered decuples of integers which, although bulkier, would be little more instructive than the ordered sextuples of septrights.

Another reason to study septrights rather than quintrights is that the latter derive some of their properties from the square root of five, which presents a famously extraordinary variety of interesting characteristics; there is consequent risk that quintrights might display properties that have no analogue with the higher orders.

For those reasons, septrights have been chosen as the vehicle for this report.

Section 1. A septright is an ordered sextuple of integers, the components of which are written, separated by commas, between shallow angle brackets. Examples are ⟨ 3, −1, −5, 6, 8, −14 ⟩ and ⟨ 0, 0, 0, 0, 0, 0 ⟩. If A is a septright, its components are denoted by the subscripts q, r, s, t, u, v. Hence this notational identity:

A ≡ ⟨ Aq, Ar, As, At, Au, Av

Another identity defines the meaning of a septright in terms of real numbers, specifically cosines; the factor of 2 greatly simplifies the multiplication table to be encountered later:

 A ≡ 2 Aq cos ( 90° / 7) + 2 Ar cos (180° / 7) + 2 As cos (270° / 7) + 2 At cos (360° / 7) + 2 Au cos (450° / 7) + 2 Av cos (540° / 7)

Equivalently:

 A ≡ 2 Aq cos (1π / 14) + 2 Ar cos (2π / 14) + 2 As cos (3π / 14) + 2 At cos (4π / 14) + 2 Au cos (5π / 14) + 2 Av cos (6π / 14)

Importantly, ⟨ 0, +1, 0, −1, 0, +1 ⟩ equals the ordinary integer 1. More broadly, any integer can be written as a septright.

Convenient is to give names to certain singleton septrights:

 Q ≡ ⟨ 1, 0, 0, 0, 0, 0 ⟩ ≈ 1.949856 R ≡ ⟨ 0, 1, 0, 0, 0, 0 ⟩ ≈ 1.801938 S ≡ ⟨ 0, 0, 1, 0, 0, 0 ⟩ ≈ 1.563663 T ≡ ⟨ 0, 0, 0, 1, 0, 0 ⟩ ≈ 1.246980 U ≡ ⟨ 0, 0, 0, 0, 1, 0 ⟩ ≈ 0.867767 V ≡ ⟨ 0, 0, 0, 0, 0, 1 ⟩ ≈ 0.445042

A septright is said to be an rtv3 if it has zeroes in the first, third, and fifth positions, as ⟨ 0, +14, 0, −3, 0, +22 ⟩. Similarly, a septright is a qsu3 if it has zeroes in the second, fourth, and sixth positions, as ⟨ +38, 0, +19, 0, −6, 0 ⟩. The all-zero septright is both, while many septrights are neither.

The septrights are patently a subset of the real numbers. Note that the integers are denumerable, and ordered sextuplets of them are also denumerable. Since the real numbers are not denumerable, there must be some reals that cannot be written as septrights. Therefore the septrights are a proper subset of the real numbers.

Septright comes from sept, meaning one-seventh, and right, meaning right angle. The angles in the formulae are all multiples of 90°/7. A possible adjective is septrighteous.

Section 2. The rules for arithmetic derive from the cosine identity above. Addition and subtraction are inevitable:

A + B = ⟨ Aq + Bq, Ar + Br, As + Bs, At + Bt, Au + Bu, Av + Bv

AB = ⟨ AqBq, ArBr, AsBs, AtBt, AuBu, AvBv

Multiplication by an integer scalar n is not a problem:

nA = An = ⟨ Aqn, Arn, Asn, Atn, Aun, Avn

Now this can be written:

AAqQ + ArR + AsS + AtT + AuU + AvV

The rule for multiplication of two septrights is lengthy, the result of a derivation (not shown here) employing many steps of elementary algebra and trigonometry. A direct procedure for calculating the product AB starts with defining 21 integer quantities which for clarity have two-letter names:

 qq = AqBq rr = ArBr ss = AsBs tt = AtBt uu = AuBu vv = AvBv
 qr = AqBr + ArBq
 qs = AqBs + AsBq rs = ArBs + AsBr
 qt = AqBt + AtBq rt = ArBt + AtBr st = AsBt + AtBs
 qu = AqBu + AuBq ru = ArBu + AuBr su = AsBu + AuBs tu = AtBu + AuBt
 qv = AqBv + AvBq rv = ArBv + AvBr sv = AsBv + AvBs tv = AtBv + AvBt uv = AuBv + AvBu

Then the six components of AB are:

 (AB)q = qr + rs + st + tu + uv (AB)r = qs + rt + su + tv + vv + 3qq + 2(rr + ss + tt + uu) (AB)s = qr + qt + ru + sv − uv (AB)t = qs + qu − rr + rv − tv − 3uu − 2(qq + ss + tt + vv) (AB)u = qt + qv + rs − sv − tu (AB)v = qu + rt − rv − su + tt + 3ss + 2(qq + rr + uu + vv)

The following table, with numerical approximations, is another way to say it:

 septrightmultiplication Q2 cos ( 90° / 7)2 sin (540° / 7)1.949856 R2 cos (180° / 7)2 sin (450° / 7)1.801938 S2 cos (270° / 7)2 sin (360° / 7)1.563663 T2 cos (360° / 7)2 sin (270° / 7)1.246980 U2 cos (450° / 7)2 sin (180° / 7)0.867767 V2 cos (540° / 7)2 sin ( 90° / 7)0.445042 Q1.949856 2 + R3.801938 Q + S3.513519 R + T3.048917 S + U2.431430 T + V1.692021 U0.867767 R1.801938 Q + S3.513519 2 + T3.246980 Q + U2.817623 R + V2.246980 S1.563663 T − V0.801938 S1.563663 R + T3.048917 Q + U2.817623 2 + V2.445042 Q1.949856 R − V1.356896 S − U0.695895 T1.246980 S + U2.431430 R + V2.246980 Q1.949856 2 − V1.554958 Q − U1.082088 R − T0.554958 U0.867767 T + V1.692021 S1.563663 R − V1.356896 Q − U1.082088 2 − T0.753020 Q − S0.386193 V0.445042 U0.867767 T − V0.801938 S − U0.695895 R − T0.554958 Q − S0.386193 2 − R0.198062 where 2 = 2R − 2T + 2V

Among the 36 products listed in table above, it so happens that all 15 operations in the upper left triangle (shaded in green) are addition; all 15 in the lower right (red), subtraction. Also:

 rtv3 × rtv3 = rtv3 qsu3 × rtv3 = qsu3 qsu3 × qsu3 = rtv3

Because septrights are real numbers, addition and multiplication are commutative and associative. The septright containing all zeroes is the additive identity, while RT + V = ⟨ 0, +1, 0, −1, 0, +1 ⟩ is the multiplicative identity. Multiplication distributes over addition, and the identity for addition is the annihilator for multiplication.

A dot product of two septrights, resembling the standard inner product of vector analysis, and notated A · B, can also be formed. Define this integer:

p = AqBq + ArBr + AsBs + AtBt + AuBu + AvBv

Septright A · B is an rtv3, its components being:

 (A · B)q = 0 (A · B)r = (AqBq − AvBv) + 2p (A · B)s = 0 (A · B)t = (ArBr − AuBu) − 2p (A · B)u = 0 (A · B)v = (AsBs − AtBt) + 2p

Although p by itself looks like the usual definition of the standard inner product, the presence of cosines in the definition of a septright adds complication.

The multiplication operation is the reason that cosines instead of sines were selected for the definition of septrights. The following standard identity uses cosines only:

2 cos x cos y = cos (xy) + cos (x + y)

In adaptation for direct use in septrights, the formula is doubled:

(2 cos x) (2 cos y) = 2 cos (xy) + 2 cos (x + y)

Two similar-looking candidate identities would have mixed sines and cosines, entailing inconvenience:

2 sin x sin y = cos (xy) − cos (x + y)
2 sin x cos y = sin (xy) + sin (x + y)

The multiplication rule for the hyperbolic cosine closely resembles that for the circular cosine:

2 cosh x cosh y = cosh (xy) + cosh (x + y)

However, it does not work due to complications.

Section 3. As would be expected with real numbers, division by zero fails. Even with a non-zero divisor, division does not work in general, because it often leads to non-integers as elements of the quotient. Hence no division ring is formed. Were the septrights defined as a rational domain (in other words, a field) rather than an integer domain, this problem would evaporate.

Examples of non-divisibility:

NOT septrights
thiswould have been
1 ÷ Q 17 (+ 3Q − 2S + 1U)
1 ÷ S 17 (− 1Q + 3S + 2U)
1 ÷ U 17 (+ 2Q + 1S + 3U)

There is a scheme that will find the quotient of A and B whenever it does exist. Let C = A ÷ B; then CB = A, and the multiplication rule can be written this way:

 Aq = Cq(Br) + Cr(Bq + Bs) + Cs(Br + Bt) + Ct(Bs + Bu) + Cu(Bt + Bv) + Cv(Bu)
 Ar = Cq(3Bq + Bs) + Cr(Bt + 2Br) + Cs(Bq + Bu + 2Bs) + Ct(Br + Bv + 2Bt) + Cu(Bs + 2Bu) + Cv(Bt + Bv)
 As = Cq(Br + Bt) + Cr(Bq + Bu) + Cs(Bv) + Ct(Bq) + Cu(Br − Bv) + Cv(Bs − Bu)
 At = Cq(Bs + Bu − 2Bq) + Cr(− Br + Bv) + Cs(Bq − 2Bs) + Ct(− Bv − 2Bt) + Cu(Bq − 3Bu) + Cv(Br − Bt − 2Bv)
 Au = Cq(Bt + Bv) + Cr(Bs) + Cs(Br − Bv) + Ct(Bq − Bu) + Cu(− Bt) + Cv(Bq − Bs)
 Av = Cq(Bu + 2Bq) + Cr(Bt − Bv + 2Br) + Cs(3Bs − Bu) + Ct(Br + Bt) + Cu(Bq − Bs + 2Bu) + Cv(− Br + 2Bv)

Regard this as a system of six linear equations in the six unknowns Cq, Cr, Cs, Ct, Cu, Cv; and solve. If all six results are integers, division was successful.

The sum of non-septrights might nonetheless be a septright, for instance Q −2 + S −2 + U −2 = 2. In a case like this, one might informally explain that there are some rational numbers behind the scenes taking care of business.

Septrights do form an integral domain, and rtv3s constitute an integral subdomain.

 subdomainmultiplication R T V R 2 + T R + V T − V T R + V 2 − V R − T V T − V R − T 2 − R

Septrights of the form ⟨ 0, +n, 0, −n, 0, +n ⟩ go further, producing the archetypal integral domain. By contrast, qsu3s do not establish an integral domain, because they are not closed under multiplication.

As an alternative to ordinary division, a quotient-and-remainder operation can be defined, as follows:

• Divide one septright by another, obtaining an exact quotient by using rational arithmetic.
• Round each rational component of the quotient to the nearest integer.
• Form the remainder by subtracting, from the dividend, the product of the divisor and the rounded quotient.
This is rather like the procedure used with ordinary integers. Example:

 dividend: ⟨ −9, 5, −4, 12, 4, −3 ⟩ ≈ 2.306034 divisor: ⟨ 2, 1, 0, −2, 1, 1 ⟩ ≈ 4.520500 exact quotient: ⟨ −1027, −3656, 4781, 653, 436, 1302 ⟩ ÷ 1289 ≈ 0.510128 rounded quotient: ⟨ −1, −3, 4, 1, 0, 1 ⟩ ≈ 0.591004 remainder: ⟨ 0, 2, −1, −3, 0, 3 ⟩ ≈ −0.365601

Not yet established is whether this method guarantees that the magnitude of the remainder will be less than the magnitude of the divisor, as would be required for a Euclidean domain. For that matter, it is not yet clear whether the magnitude, for this purpose, ought to be defined as the magnitude of the septright as a real number; or the sum of the squares of the components without regard to the cosines; or something else.

Section 4. Because septrights are based on trigonometry, many identities can be written.

Identities involving multiplication
 Q 2 + V 2 = 4 R 2 + U 2 = 4 S 2 + T 2 = 4
 Q 2 − V 2 = 2R R 2 − U 2 = 2T S 2 − T 2 = 2V
 RTU = Q QRV = S STV = U RTV = + R − T + V = 1
 QRT = + Q + S + U = Q ÷ V = tan (540° / 7) TUV = − Q + S + U = U ÷ R = tan (180° / 7) RSV = + Q − S + U = S ÷ T = tan (360° / 7) QSU = + Q + S − U = √7
 QR · SV = 2 + V SV · TU = 2 − T TU · QR = 2 + R
 QT · RS = + 4R − 1T + 2V RS · UV = + 1R − 2T + 4V UV · QT = − 2R + 4T − 1V

The identities RTV = + RT + V and QSU = + Q + SU exemplify an often-seen phenomenon: T will be of a sign opposite that of R and V; U opposite Q and S.

Identities involving division
 R ÷ T = V + 1 T ÷ V = R + 1 V ÷ R = T − 1
 Q ÷ R = + Q − U S ÷ V = + Q + S U ÷ T = + S − U
 R ÷ V = + 2R + 1V T ÷ R = − 1R + 2T V ÷ T = + 1T − 2V
 Q ÷ V = + Q + S + U S ÷ T = + Q − S + U U ÷ R = − Q + S + U
 7R ÷ U = + 3Q + 5S + 1U 7T ÷ S = + 5Q − 1S − 3U 7V ÷ Q = + 1Q − 3S + 5U
 7R ÷ Q = + 1Q + 4S − 2U 7T ÷ U = + 4Q + 2S − 1U 7V ÷ S = + 2Q + 1S − 4U
 Q ÷ U = 1 ÷ V = + R + V S ÷ Q = 1 ÷ T = + T − V U ÷ S = 1 ÷ R = + R − T
 7R ÷ S = 7 ÷ U = + 2Q + 1S + 3U 7T ÷ Q = 7 ÷ S = − 1Q + 3S + 2U 7V ÷ U = 7 ÷ Q = + 3Q − 2S + 1U

Patterns formed by the identities lead to the defining of triads. Triads are a good example of the advantage of studying septrights instead of quintrights, as the analogous dyads of quintrights would probably not have been conspicuous enough to be noticed.

There are many sequences of identities that yield either integers or integral multiples of √7.

Section 5. Unknown to the author is an exact method of determining whether one septright is greater than, or less than, another. The same considerations apply to detecting if two septrights with different components are nonetheless equal. Nearly all computers offer a floating-point approximation which is usually satisfactory, but which will be indefinite in close cases because roundoff error may dominate.

With quintrights, the problem is manageable because their values are defined by relatively simple expressions involving the square roots of real numbers, and a lengthy but feasible brute-force algebraic procedure provides the answer. On the other hand, the corresponding expressions for septrights become intractable because of the presence of cube roots and complex numbers, even though the results are real numbers.

Section 6. The six singletons Q, R, S, T, U, V can be rearranged and negated in five alternate ways to satisfy the generic septright multiplication table below:

 genericmultiplication q r s t u v q 3r − 2t + 2v q + s r + t s + u t + v u r q + s 2r − t + 2v q + u r + v s t − v s r + t q + u 2r − 2t + 3v q r − v s − u t s + u r + v q 2r − 2t + v q − u r − t u t + v s r − v q − u 2r − 3t + 2v q − s v u t − v s − u r − t q − s r − 2t + 2v

Values from any one column below will make it work:

 orig. alt. 1 alt. 2 alt. 3 alt. 4 alt. 5 q = +Q −Q +U −U +S −S r = +R +R −T −T +V +V s = +S −S −Q +Q −U +U t = +T +T −V −V −R −R u = +U −U +S −S −Q +Q v = +V +V +R +R −T −T

A trivial solution arises from solving these simultaneous equations:

 rt = r + v rv = t − v tv = r − t

Produced is r = t = v = 0, whence q = s = u = 0.

Also appearing is the partial solution r = − t = v = −2. However, no values of q, s, and u are simultaneously consistent with this.

Beyond all that, there are solutions in integer matrices.

Section 7. Suppose that a, b, c and d are ordinary integers, and n is a positive integer. Then a basic result from modular arithmetic is that if both of these are true:

• a = b [mod n]
• c = d [mod n]
then all three of these will be true:
• a + c = b + d [mod n]
• ac = bd [mod n]
• a × c = b × d [mod n]

Now define modulo equality for septrights: A = B [mod n] if and only if all six of these are true:

• Aq = Bq [mod n]
• Ar = Br [mod n]
• As = Bs [mod n]
• At = Bt [mod n]
• Au = Bu [mod n]
• Av = Bv [mod n]

It is easy to prove that if both of these are true:

• A = B [mod n]
• C = D [mod n]
then all three of these will be true:
• A + C = B + D [mod n]
• AC = BD [mod n]
• A × C = B × D [mod n]

Part II,
in which the singleton letters sometimes stand for merely the associated rational numbers.

Section 8. MathWorld cites an "interesting" result from Borwein and Bailey:

∛cos (2π / 7) + ∛cos (4π / 7) + ∛cos (6π / 7) = − ∛[12 (3∛7 − 5)]

Recall that T / 2 = cos (2π / 7), − V / 2 = cos (4π / 7), and R / 2 = cos (6π / 7). Substitution and simplification yields this:

R − ∛T + ∛V = ∛(3∛7 − 5)

Now cube both sides and simplify:

− ∛(RRT) + ∛(RRV) + ∛(TTR) + ∛(TTV) + ∛(VVR) − ∛(VVT) = ∛7

In an elementary but lengthy procedure, cube both sides again, simplify, and rearrange. Then define these six constants:

 A = + 1R − 5T + 5V B = + 5R − 1T + 5V C = − 5R + 5T − 1V D = − 2R + 5T − 4V E = + 4R − 2T + 5V F = − 5R + 4T − 2V

Now the following can be written:

A∛(TTV) + B∛(VVR) + C∛(RRT) = D∛(TTR) + E∛(VVT) + F∛(RRV)

For a third time, cube both sides of the equation. After simplification, there will be no more roots to extract, and it can be seen that both sides of the equation are indeed equal, in particular equal to zero, verifying Borwein and Bailey.

Displaying triad behavior, A + BC = +11 and DE + F = −11. Also integers are ABC = +71 and DEF = −29.

Section 9. The following results come from Witula et alii, but here have been written using the septright singleton letters, which sometimes afford a more concise notation.

R · ∛(R ÷ T) − V · ∛(V ÷ R) − T · ∛(T ÷ V) = 0

∛(1 + R) + ∛(1 − T) + ∛(1 + V) = ∛(R ÷ T) − ∛(V ÷ R) + ∛(T ÷ V) = ∛7

∛(34 − S · √567) + ∛(34 − Q · √567) + ∛(34 + U · √567) = 0

∛(SU(Q − √7)) + ∛(UQ(S − √7)) + ∛(QS(U + √7)) = 0

∛(3R − 5T + 2V) + ∛(2R − 3T + 5V) + ∛(5R − 2T + 3V) = ∛49

Note that the radicands in the last identity form an rtv3 triad whose sum is 10 and product is 1.

Section 10. Values R, −T, and V solve the equation x3x2 − 2x + 1 = 0. They can be written in a non-trigonometric form, as follows.

First define A1 = −754 and B1 = −49108. Then choose:

 Method 1. Define the complex number C1 = ∛(A1 + √B1). Either of the square roots, and any one of the three cube roots, may be selected; the same answers will result. The roots of the polynomial are: 1⁄3 + 2 Re(C1) 1⁄3 − Re(C1) + √3 Im (C1) 1⁄3 − Re(C1) − √3 Im (C1) These are R, −T, and V; but which is which depends on the choice of square and cube roots in the calculation of C. All six permutations are possible. Method 2. Define the complex numbers D1 = ∛(A1 + √B1) and E1 = ∛(A1 − √B1). The square and cube roots must be selected so that D1E1 = 7⁄9; this may require a little trial and error. The roots of the polynomial are: 1⁄3 + (D1 + E1) 1⁄3 − 1⁄2 (D1 + E1) + i⁄2 √3 (D1 − E1) 1⁄3 − 1⁄2 (D1 + E1) − i⁄2 √3 (D1 − E1)

Similarly, values Q, S, and −U solve the equation x3 − √7x2 + √7 = 0. They too can be written in a non-trigonometric form.

First define A2 = −1354 √7 and B2 = −7108. Then choose:

 Method 1. Define the complex number C2 = ∛(A2 + √B2). As with R-T-V method 1, any roots may be selected. The roots of the polynomial are: 1⁄3 √7 + 2 Re(C2) 1⁄3 √7 − Re(C2) + √3 Im (C2) 1⁄3 √7 − Re(C2) − √3 Im (C2) Method 2. Define the complex numbers D2 = ∛(A2 + √B2) and E2 = ∛(A2 − √B2). The square and cube roots must be selected so that D2E2 = 7⁄9. The roots of the polynomial are: 1⁄3 √7 + (D2 + E2) 1⁄3 √7 − 1⁄2 (D2 + E2) + i⁄2 √3 (D2 − E2) 1⁄3 √7 − 1⁄2 (D2 + E2) − i⁄2 √3 (D2 − E2)

Another polynomial of interest.