### Avoided Numbers

May 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.

While

seven is considered to be a

lucky number by some people, it was a number to be avoided in certain

cultures under certain circumstances. When doubly

cursed by seven, as being the

seventh son of a seventh son, you might be a

vampire. Fortunately,

science has dispelled such

numerology, although

some physicists think that

it still persists in

string theory.[2]

Consider a person who's so afraid of the number, seven, that he refuses to leave his

house on the seventh day of each

month. Moreover, he chooses to remain inside on the 17th and 27th of each month. A little strange, but he could still function in the

modern world by making up lost

work days on

weekends.

Then comes the month of

July, the seventh month. He would need to stock a lot of

food, and take that month as his work

vacation. He's still able to function in our modern world. His big problem arises next year,

2017. While we leave our subject's

researching use of the

Hebrew calendar and other

calendars as a possible remedy, it's interesting to consider how many numbers we would need to reject from the

natural numbers if we wished to avoid a particular

digit.

Often it's easier to

calculate something in the

converse, so we'll calculate the

probability that a number

*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.

| *Probability that a number will not contain a particular digit (for example, seven) as a function of the number of its digits.*
(Graphed using Gnumeric.) |

This result is simply expressed as

**p(n) = (9/10)**^{n}, where

**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)**^{n}. 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.

To find how many numbers fulfill our avoided-digit

criterion up to a number

**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,

*O*(N^{(1-c)}), where

**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.

While the details of Maynard's proof in his 44 page paper are beyond my grasp, he apparently was able to show that there are infinitely many prime numbers which do not have a particular digit in their decimal expansion.[3]

### 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.