Binary Representations of Inverted Primes

I have been reading a book by Manfred Schroeder called “Fractals, Chaos, Power Laws” and in it he uses binary numbers to explore a variety of patterns, including some involving the Fibonacci numbers, the golden mean, etc. This got me thinking about trying to express the primes in binary format to look for patterns. I took the inverse of the first 11 primes (to 31) and converted them to binary format and obtained the following:

If you study these carefully, you will see a couple of interesting things:

– excepting the special case of P1 = 2, the number of repeating digits for the n-th prime, Pn, is (Pn-1)
– the repeating groups of digits in each case have symmetry, in that the second half of the group is either identical to the first half, or, more commonly in these sets, is the inverse of the first half (e.g. 11101 in place of 00010)

I don’t know if these observations have any particular use or significance, but I thought they were interesting enough to note here for posterity. 🙂


About petequinn

I'm a Canadian geotechnical engineer specializing in the study of landslides. I started this page to discuss some mathematical topics that interest me, initially this involved mostly prime numbers, but more recently I've diverted focus back to a number of topics of interest in geotechnique, geographic information systems and risk. I completed undergraduate training in engineering physics at Royal Military College (Kingston, Ontario), did a masters degree in civil (geotechnical) engineering at University of British Columbia (Vancouver), and doctorate in geological engineering at Queen's University (Kingston). I was a military engineer for several years at the beginning of my career, and did design and construction work across Canada and abroad. I've worked a few years for the federal government managing large environmental clean up projects in Canada's arctic, and I've worked across Canada, on both coasts and in the middle, as a consulting geotechnical engineer. My work has taken me everywhere in Canada's north, to most major Canadian cities and many small Canadian towns, and to Alaska, Chile, Bermuda, the Caribbean, Germany, Norway, Sweden, Bosnia, and Croatia. My main "hobby" is competitive distance running, which I may write about in future.
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