[seqfan] Project: sequences obtained from Gaussian Integers via Hungarian binary method of encoding?
Antti Karttunen
antti.karttunen at gmail.com
Sat Aug 20 21:31:00 CEST 2016
Here's an old idea from my attic:
Consider the method of expressing Gaussian Integers as sums of powers of
(-1 + i) (with coefficients just either 0 or 1).
I call this "Hungarian binary method", as I first saw it at this web page:
http://szdg.lpds.sztaki.hu/szdg/desc_numsys_es.php
(but please tell if it is much older and discovered in some other country!)
By hand calculation, I get the following values for some small binary
vectors:
0000 = 0
0001 = 1
0010 = -1 + i
0011 = i
0100 = -2i
0101 = 1 - 2i
0110 = -1 - i
0111 = -i
1000 = 2 + 2i = (-1+i)^3
1001 = 3 + 2i
1010 = 1 + 3i
1011 = 2 + 3i
1100 = 2
1101 = 3
1110 = 1 + i
1111 = 2 + i
10000 = (-1+i)^4 = 1 + -4i + -6 + 4i + 1 = 2 -6 = -4
so with these we finally get:
11101 = -1 (for a moment I was afraid that it would not appear at all!)
Would it make sense to express with this system all kind of functions whose
domains and/or ranges are Gaussian Integers?
(As OEIS sequences, I mean).
That is, in a slightly similar way as functions whose domain and/or range
are polynomials over GF(2) have been submitted via their encoded binary
representation, found under index entry http://oeis.org/wiki/Index_to_
OEIS:_Section_Ge#GF2X
Or are there better ones for Gaussian Integers? I'm not thinking here about
how easy it is for human beings to search the terms, but in terms of some
vaguely felt elegance/inelegance of the sequences encoded with it. That -1
appears so "late" (encoded by binary representation of 29) in this system
feels a bit worrisome.
In any case, as long as bitwise-AND(x,y) = 0 (x and y have no 1-bits in
common positions when encoded), then Gaussian_Integer_interpretation(x+y) =
Gaussian_Integer_interpretation(x) + Gaussian_Integer_interpretation(y).
For the starters, could somebody generate a list of numbers whose binary
representations map via this particular encoding to Gaussian primes? (I
don't have appropriate mathematical software at my reach, or cannot use it
properly).
Best regards,
Antti
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