S-expression: Difference between revisions
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m →Mathematical derivation: finished proofs (needs proofchecking) |
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This result will be useful, so we will refer to it as [Eq. 1]: | This result will be useful, so we will refer to it as [Eq. 1]: | ||
'''[Eq. 1]''' | '''[Eq. 1]''' Sk/S(k+2) = (k<sup>3</sup> + 3k<sup>2</sup>)/( k<sup>3</sup> + 3k<sup>2</sup> - 4 ) | ||
Note that when k = 2n in [Eq. 1], everything in the numerator and denominator is divisible by 4 because the only instances of k have it raised to a power of 2 or greater meaning there will be a factor of (2n)<sup>2</sup> = 4n<sup>2</sup>, therefore Sk/S(k+2) is superparticular when k is even. | Note that when k = 2n in [Eq. 1], everything in the numerator and denominator is divisible by 4 because the only instances of k have it raised to a power of 2 or greater meaning there will be a factor of (2n)<sup>2</sup> = 4n<sup>2</sup>, therefore Sk/S(k+2) is superparticular when k is even. | ||
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= ( 4m + 4(3a) + 4 )/( 4m + 4(3a) ) = ( m + 3a + 1 )/( m + 3a ) | = ( 4m + 4(3a) + 4 )/( 4m + 4(3a) ) = ( m + 3a + 1 )/( m + 3a ) | ||
Then for the final case we want to show that S(4n | Then for the final case we want to show that S(4n-1)/S(4n+1) is odd-particular by setting k=4n-1 in [Eq. 1]: | ||
S(4n-1)/S(4n+1) = ( (4n-1)<sup>3</sup> + 3(4n-1)<sup>2</sup> )/( (4n-1)<sup>3</sup> + 3(4n-1)<sup>2</sup> - 4 ) | |||
As before we make replacements: | |||
(4n-1)<sup>3</sup> = (4n)<sup>3</sup> - 3*(4n)<sup>2</sup> + 3*4n - 1 is of the form 4m-1. | |||
(4n+1)<sup>2</sup> = (4n)<sup>2</sup> - 2*4n + 1 is of the form 4a+1. | |||
Implying: S(4n-1)/S(4n+1) = ( 4m-1 + 3(4a+1) )/( 4m-1 + 3(4a+1) - 4 ) | |||
= ( 4m-1 + 4(3a) + 3 )/( 4m-1 + 4(3a) + 3 - 4 ) | |||
= ( 4m + 4(3a) + 2 )/( 4m + 4(3a) - 2 ) = ( 2m + 2(3a) + 1 )/( 2m + 2(3a) - 1) | |||
...which is of the form (x + 1)/(x - 1) meaning it is odd-particular. | |||
In conclusion: Sk/S(k+2) is superparticular for all integer 2 < k =/= 3 (mod 4) and is odd-particular when k = 3 (mod 4). | |||
[[Category:Math]] | [[Category:Math]] | ||