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Side dishes Math On Root Approach Carréphobic numbers with carréphylic ancestors or descendants
Side dishes Math On Root Approach Carréphobic numbers with carréphylic ancestors or descendants
| Carréphobic numbers with carréphylic ancestors or descendants |
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Carréphylic ancestors Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that x = a2y ⇔ √x = a√y. Since y is carréphylic, the approach of √x should pose no practical problems. Here's how to find the 's' and 'p' of the rational approach of x: s2-xp2 = 1 ⇔ s2-y(ap)2 = 1. The fraction s/ap is the ath fraction of the set of sp-blocks of the rational approach of y, starting from the trivial block. It is therefore the first fraction of the base-a series accelleration of the rational approach of y. The corresponding factor '2s' is Fa or the factor of y 'base-a'. Since √x = a√y the sp-fraction of x is a(s/ap) = s/p. Example carréphylic ancestry The position of a fraction indicates whether it is over or under the root-value. The approach of √7 - carréphylic
The approach of √28=2√7 - carréphobic
Using the above data we find that the first non trivial sp-fraction of 28 is twice the second non-trivial fraction of 7, or 127/24. The corresponding factor 2*127 is the base-2 accelleration of the factor of 7: 162-2. The approach of √63=3√7 - carréphylic
63 is one less than a square, so the exception mentioned in on root approach applies: 127 and 16, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 8 and 1 because 82-63*12 = 1 satisfies the diophantine equation. Nevertheless, trice the third non-trivial fraction of 7 is 2024/255 and the resulting series is the base-3 accelleration of the sp-series of √63. The corresponding factor 2*2024 is the base-3 accelleration of the factor of 7: 163-3*16. The approach of √112=4√7 - carréphobic As it happens 112 falls outside the first 100 non-squares, so I didn't do a search, but using the above data the first non trivial sp-fraction of 112 is four times the fourth non-trivial fraction of 7, or 32257/3048. The corresponding factor 2*32257 is the base-4 accelleration of the factor of 7: 164-4*162+2. Carréphylic descendants Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that a2x = y ⇔ a√x = √y. Here's how to find the 's' and 'p' of the rational approach of x: For y the first non-trivial 's' and 'p' satisfy: s2-yp2 = 1 ⇔ s2-x(ap)2 = 1 ⇔ 's' and 'ap' are the first non-trivial element of the set of sp-blocks of the rational approach of x. The corresponding factor is the same. Example carréphylic descendency The approach of √13 - carréphobic The fractional approach of root 13, the first carréphobic number, as provided by Ed's straightforward program, looks like this:
As it happens, 52*13 = 325 = 182-1 and thus carréphylic. The formula renders 649/36 as the first sp-block of the approach of 5√13, making 649/180 the first sp-block of √13 with the same factor 1298. Easy as fruitpie. |
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