Complex Base 1
Public  08/01/15  15xp  Math  52.6% 
A complex integer is a number $ p+q \times i \: (i^2 = 1) $ where p and q are integers.
It can be proved that every complex integer has a unique representation in base $ i1 $ with 'digits' 0 or 1.
For instance, we have:
$ 2 = 0\times{(1+i)}^0+0\times{(1+i)}^1+1\times{(1+i)}^2+1\times{(1+i)}^3 \; // 1 1 0 0 $ $ 6 = 0\times{(1+i)}^0+0\times{(1+i)}^1+1\times{(1+i)}^2+1\times{(1+i)}^3+1\times{(1+i)}^4+ \dots $ $ \qquad \dots +0\times{(1+i)}^5+1\times{(1+i)}^6+1\times{(1+i)}^7+1\times{(1+i)}^8 \; // 1 1 1 0 1 1 1 0 0 $
Actually, 6 is the smallest natural integer whose representation contains 6 digits to '1'
Let's define the function R(n) for an natural integer n as follow: consider the digits of n in base $ i1 $ as the binary representation of an integer.
Thus:
R(2) = 12
R(6) = 476
Find the smallest integer a, whose representation contains 60 digits
Find the smallest integer b, whose representation contains 93 digits to '1'
Answer format: a,R(a),b,R(b)
[My timing: instant]
It can be proved that every complex integer has a unique representation in base $ i1 $ with 'digits' 0 or 1.
For instance, we have:
$ 2 = 0\times{(1+i)}^0+0\times{(1+i)}^1+1\times{(1+i)}^2+1\times{(1+i)}^3 \; // 1 1 0 0 $ $ 6 = 0\times{(1+i)}^0+0\times{(1+i)}^1+1\times{(1+i)}^2+1\times{(1+i)}^3+1\times{(1+i)}^4+ \dots $ $ \qquad \dots +0\times{(1+i)}^5+1\times{(1+i)}^6+1\times{(1+i)}^7+1\times{(1+i)}^8 \; // 1 1 1 0 1 1 1 0 0 $
Actually, 6 is the smallest natural integer whose representation contains 6 digits to '1'
Let's define the function R(n) for an natural integer n as follow: consider the digits of n in base $ i1 $ as the binary representation of an integer.
Thus:
R(2) = 12
R(6) = 476
Find the smallest integer a, whose representation contains 60 digits
Find the smallest integer b, whose representation contains 93 digits to '1'
Answer format: a,R(a),b,R(b)
[My timing: instant]
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