## Avoided NumbersMay 12, 2016 New York State initiated its sales tax on April 14, 1965, while I was there attending high school. At its initiation, this tax was just 2%, but it quickly ramped up over the years, with counties adding their own sales taxes. Fortunately, many items are excluded from sales tax, including funeral expenses. In a small sense, you can take it with you. When this sales tax was initiated, I realized that it would be impossible to buy some items at certain prices. As an example, consider the sales tax for New Jersey, presently at 7%.[1] There is no sales tax on an item costing ten cents, but there's a one cent sales tax on an item costing eleven cents. That means that you can't pay eleven cents for an item. You can only buy that item for twelve cents. Eleven is a forbidden number. Because of this sales tax, it's impossible to buy items for 11, 21, 35, 51, 67, 83, or 97 cents. As the following graph shows, there are seven forbidden prices in each dollar interval.
does not contain a particular digit. For numbers from 0-9, this would be 9/10. For a two digit number, we need to multiply this probability to get (81/100), which means we should expect 19 numbers in that range to contain a seven, which is verified by inspection. As the following graph shows, it becomes more difficult to find a number devoid of the digit, seven, as we go to larger numbers.
p(n) = (9/10), where ^{n}n is the number of digits of the number, and it can be expressed in an arbitrary base b as p(n) = (b-1/b). Obviously, the curve for hexadecimal numbers (base-16) will descend less quickly, and octal numbers (base-8) will have a steeper curve, as shown below.
^{n}
N, we need to integrate under these curves. Lazy mathematician that I am, I resort to the result given in a recent arXiv paper by James Maynard.[3]
The numbers fulfilling the avoided-digit criterion up to N are of the order, , where O(N^{(1-c)})c = log(b/(b-1))/log(b). For decimal numbers, c ≈ 0.045757. Although the probability is low, up to 100 decimal digits we still obtain 2.65 x 10^{95} numbers meeting our avoided digit criterion.
In his arXiv paper, Maynard addressed the question of whether there are an infinity of prime numbers in the set of such avoided-digit numbers. The infinity of primes among all the natural numbers was established by Euclid in his Elements, Book IX, Proposition 20, a proof that's easy enough for a high school student to understand.
## References:- Sales Tax Collection Schedule, State of New Jersey, Department of the Treasury, Division of Taxation, July 15, 2006 (PDF file).
- Lee Smolin, "The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next," Mariner Books, September 4, 2007, 416 pp, ISBN: 978-0618918683.
- James Maynard, "Primes with restricted digits," arXiv, April 4, 2016.
Linked Keywords: New York State; sales tax; high school; county; funeral; You Can't Take It with You; you can take it with you; price; sales tax for New Jersey; cent; forbidden; number; Cartesian coordinate system; graph; dolla; Gnumeric; seven; luck; lucky; culture; curse; seventh son of a seventh son; vampire; science; numerology; Lee Smolin; physicist; The Trouble with Physics; string theory; house; month; contemporary history; modern world; workweek and weekend; work day; weekend; July; food; vacation; 2017; research; Hebrew calendar; calendar; natural number; numerical digit; calculation; calculate; converse; probability; multiplication; multiply; verified by inspection; probability; Gnumeric; radix; base; hexadecimal; octal; curve; base-16; base-10; base-8; criterion; integral; integrate; mathematician; arXiv; James Maynard; infinite set; infinity; prime number; set; Euclid's theorem; Euclid's Elements; mathematical proof; Oxyrhynchus papyrus, I-29; Wikimedia Commons; Lee Smolin, "The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next," Mariner Books, September 4, 2007, 416 pp, ISBN: 978-0618918683. |
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