S-expression: Difference between revisions

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 When k = 4n+1, we have to do some work to show it is superparticular:
+First let's note (4n+1)<sup>3</sup> = (4n+1)(4n+1)(4n+1) = (4n)<sup>3</sup> + 3*4(4n)<sup>2</sup> + 3*(4n)<sup>2</sup> + 1 is of the form 4m+1.
+Next let's note (4n+1)<sup>2</sup> = (4n)<sup>2</sup> + 4(4n) + 1 is also of the form 4m+1.
+Therefore we can replace their occurrences with 4m+1 and 4a+1 respectively, without having to worry about what m and a are (as we know they are positive integers).
+S(4n+1)/S(4n+3) = ( (4n+1)<sup>3</sup> + 3(4n+1)<sup>2</sup> )/( (4n+1)<sup>3</sup> + 3(4n+1)<sup>2</sup> - 4 )
+= ( 4m+1 + 3(4a+1) )/( 4m+1 + 3(4a+1) - 4)
+= ( 4m + 4(3a) + 4 )/( 4m + 4(3a) ) = ( m + 3a + 1 )/( m + 3a )
+Then for the final case we want to show that S(4n+1)/S(4n-1) is odd-particular:
 To be continued...