Home page.
Many of the identities on the home page suggest defining the septright triad, which comes in two varieties: one for rtv3s and one for qsu3s. Let a, b, c, d, e be integers, and:
- [rtv3 triad] Suppose there are three septrights of the following structure:
X′ = 〈 0, ±a, 0, ±b, 0, ±c 〉
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Y′ = 〈 0, ±b, 0, ±c, 0, ±a 〉
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Z′ = 〈 0, ±c, 0, ±a, 0, ±b 〉
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The expression ± X′ ± Y′ ± Z′ represents 8 = 23 possible results. If at least one of those results is of the form
〈 0, +d, 0, −d, 0, +d 〉
then X′, Y′, Z′ are defined to form an rtv3 triad.
- [qsu3 triad] Suppose there are three septrights of the following structure:
X″ = 〈 ±a, 0, ±b, 0, ±c, 0 〉
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Y″ = 〈 ±b, 0, ±c, 0, ±a, 0 〉
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Z″ = 〈 ±c, 0, ±a, 0, ±b, 0 〉
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The expression ± X″ ± Y″ ± Z″ represents 8 possible results. If at least one of those results is of the form
〈 +e, 0, +e, 0, −e, 0 〉
then X″, Y″, Z″ are defined to form a qsu3 triad.
If X, Y, Z form a triad, so do these:
- −X, Y, Z
- X, −Y, Z
- X, Y, −Z
If most of the signs in the expression ± X ± Y ± Z are minus, it is canonical to negate all three of X, Y, Z.
Below are examples of triads, with addition or subtraction indicated.
QR | · SV | = | + 2R − 2T + 3V | | add
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SV | · TU | = | + 2R − 3T + 2V | | add
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TU | · QR | = | + 3R − 2T + 2V | | add
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| | | + 7R − 7T + 7V | | result
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7R | ÷ U | = | + 3Q + 5S + 1U | | add
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7T | ÷ S | = | + 5Q − 1S − 3U | | add
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7V | ÷ Q | = | + 1Q − 3S + 5U | | sub
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| | | + 7Q + 7S − 7U | | result
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QT | · RS | = | + 4R − 1T + 2V | | add
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RS | · UV | = | + 1R − 2T + 4V | | add
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UV | · QT | = | − 2R + 4T − 1V | | sub
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| | | + 7R − 7T + 7V | | result
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7R | ÷ Q | = | + 1Q + 4S − 2U | | add
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7T | ÷ U | = | + 4Q + 2S − 1U | | add
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7V | ÷ S | = | + 2Q + 1S − 4U | | add
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| | | + 7Q + 7S − 7U | | result
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Q | ÷ U | = 1 ÷ V | = | + R | | + V | | add
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U | ÷ S | = 1 ÷ R | = | + R | − T | | | add
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S | ÷ Q | = 1 ÷ T | = | | + T | − V | | sub
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| | | | + 2R | − 2T | + 2V | | result
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Q | ÷ R | = | + Q | | − U | | add
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S | ÷ V | = | + Q | + S | | | add
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U | ÷ T | = | | + S | − U | | add
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| | | + 2Q | + 2S | − 2U | | result
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T | ÷ R | = | − 1R | + 2T | | | add
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V | ÷ T | = | | + 1T | − 2V | | add
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R | ÷ V | = | + 2R | | + 1V | | sub
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| | | − 3R | + 3T | − 3V | | result
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7T | ÷ Q | = 7 ÷ S | = − 1Q + 3S + 2U | | add
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7V | ÷ U | = 7 ÷ Q | = + 3Q − 2S + 1U | | add
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7R | ÷ S | = 7 ÷ U | = + 2Q + 1S + 3U | | sub
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| | | zero | | result
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R | ÷ T | = | + 1R − 1T + 2V | | add
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T | ÷ V | = | + 2R − 1T + 1V | | add
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V | ÷ R | = | − 1R + 2T − 1V | | sub
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| | | + 4R − 4T + 4V | | result
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S | ÷ T | = | + Q − S + U | | add
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U | ÷ R | = | − Q + S + U | | add
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Q | ÷ V | = | + Q + S + U | | sub
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| | | − Q − S + U | | result
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A | = | + | 1R − | 5T + | 5V | | add
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B | = | + | 5R − | 1T + | 5V | | add
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C | = | − | 5R + | 5T − | 1V | | sub
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| | + | 11R − | 11T + | 11V | | result
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D | = | − | 2R + | 5T − | 4V | | add
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F | = | − | 5R + | 4T − | 2V | | add
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E | = | + | 4R − | 2T + | 5V | | sub
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| | − | 11R + | 11T − | 11V | | result
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The triads in the bottom row come from section 8 of the home page.
Besides the sum of the elements of a triad, something might be said about their product.
Consider an rtv3 triad in the following form:
X′ = 〈 0, +a, 0, −b, 0, +c 〉
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Y′ = 〈 0, +b, 0, −c, 0, +a 〉
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Z′ = 〈 0, +c, 0, −a, 0, +b 〉
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Clearly, X′ + Y′ + Z′ = a + b + c. Further, X′Y′Z′ = d where d is a ten-term cubic polynomial:
d | = | abc
− a3
− b3
− c3
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| −
| 3a2b
− 3b2c
− 3c2a
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| +
| 4a2c
+ 4b2a
+ 4c2b
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Now consider a qsu3 triad in the following form:
X″ = 〈 +a, 0, −b, 0, +c, 0 〉
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Y″ = 〈 +b, 0, −c, 0, +a, 0 〉
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Z″ = 〈 +c, 0, −a, 0, +b, 0 〉
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Of course, X″ + Y″ + Z″ = (a + b + c)√7. Also, X″Y″Z″ = e√7 where e is:
e | = | abc
− a3
− b3
− c3
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| +
| 2a2b
+ 2b2c
+ 2c2a
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| +
| a2c
+ b2a
+ c2b
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