# Trotter Numbers & Trotter Primes

Recently (June 2001) I became aware of an interesting website, dedicated to the discovery and reporting of appearances of the number 47 in our world. It is called, appropriately enough, the 47 Society. They post e-mail notes from the members about any trivia related to what they claim is the “quintessential random number”. Well, if you have read the pages of WTM, it should come as no surprise to you that I “enlisted” in the society. And on June 8, I wrote my first e-mail to them, which said:

Hey, I like your neat project about 47. While I’m not quite ready to believe that 47 is the only number worth looking for, :>) , I do enjoy looking for number facts of any kind. So here’s my humble contribution…

About 9 years ago I wrote a letter that was published in the Nov. ’92 issue of the MATHEMATICS TEACHER (NCTM) about “1992”. You see, 1992 = 8 x 3 x 83. But also in the past thousand years only 2 other years had that same structure: a x b x ‘ab’. They were 1533 = 7 x 3 x 73 and (ta-dah!) 1316 = 4 x 7 x 47.

This morning as I lay in bed thinking about “47” (yes, this is true), it struck me that “47” was the concatenation of “4” [a square; I like squares, too] and “7” [the “ubiquitous 7″, as I like to say]. So I began examining other such cases.

We get 17, 47, 97, 167, 257, 367, … all primes so far. [But of course, 497 isn’t prime, but that was sorta to be expected.] Some future terms from here on are primes, while others are not.

BUT 47 is the 2nd prime in this sequence, and 2 is the only even prime. So that might count for something, huh? [Which brings to mind this quote: All primes are odd except 2–which is therefore the oddest of them all. [Knuth] ]

I hope you like this, and I’ll keep my eye out for more 47’s, okay?

That little comment about the sequence of numbers containing the number 47 was the inspiration of all that follows in this article. I warn you — it gets wild at times. Enjoy.

## Definition

The set of Trotter Numbers is a subset of the natural numbers, or positive integers, defined by the following rule:

T(n) = 10 * n2 + 7, where n = 1, 2, 3, …

The sequence begins: 17, 47, 97, 167, 257, 367, 497, …

Whenever a given T(n) [aka TN] is prime, it shall be called a Trotter Prime (TP).

After a few moments of close observation and reflection, one should notice that while the first six consecutive TN’s given are prime, the 7th one, 497, is a composite number. It is equal to 7 × 71. This characteristic alone, that in the TN sequence — unlike the sequence of natural numbers — primes can be consecutive, makes the set of Trotter Numbers interesting.

However, while it is quite easy to expect that among the infinite number of TN’s, some will be prime and some will be composite, you still have to test each TN to see if it is also a TP.

Lucky for us: we can easily prove that every 7th one can not be prime. Hence, there can never be a string of TP’s longer than 6. The question merely remains: do strings of 5, 4, or 3 TP’s exist? If so, where are they? That is what you, as the great prime hunter, must do — find ’em, and tag ’em!

This is where we stop on this page. Now it’s up to you. You must start investigating the topic of Trotter Numbers and Trotter Primes before you turn the page, as it were, to see what patterns or odd tid-bits of trivia you can find, then compare your results with what WTM has discovered so far.

When you finish with your reseach, collect all your notes together, and turn to Page 2. Thank you and good luck!

P.S. WTM is rather pleased to state that the sequence given above can be found in Sloane’s On-line Encyclopedia of Integer Sequences and has its own reference number: A061722.

# Trotter Dates

Naming numbers according to some property or characteristic that they possess is a legitimate — albeit egotistic — activity of some mathematicians. (See Keith Numbers for a simple and elegant example.)

Another interesting name category that I have recently learned about is called “Niven Numbers”. These are merely numbers that are divisible by the sum of the digits of the number. [Or to put it another way: the sum of the digits is a factor of the number itself.] A simple example should suffice for our purposes here. 126 is a Niven number because 1 + 2 + 6 = 9, and 126 divided by 9 is 14.

So I decided to combine the idea of “Product Dates” (the date part) with that of Niven numbers (the divisibility part) and came up with a new category of numbers, or dates, for all my Trotter Math friends of the world: TROTTER DATES.

A “Trotter Date” shall be defined to be a date for which the year number — either the short 2-digit form or the full 4-digit form — is divisible by the sum of the month number and the day number.

Let’s take an example, my own birthday: May 21. In 1978 we would have written this as 5/21/78. And 78 is divisible by 26, the sum of 5 and 21. Hence, that date shall be considered as a “Trotter Date”.

Of course, my birthday produces two other TD’s (Trotter Dates) in a given century, namely 5/21/26 (I wasn’t alive for that one!) and 5/21/52 (I was alive for that one, but was not aware of its signifigence at the time.)

Now if we go for the full 4-digit form using my birthday, there are some other TD’s awaiting us. For example, 5/21/1976 was my latest personal TD because 1976 divided by 26 is 76! (Hmm…, now that’s a nice coincidence.)

My next one will occur on 5/21/2002 because,… well, you understand this by now don’t you? (By the way, that date will be significant for me and my family; 2002 is the year my son graduates from high school!)
Searching for more TD’s now can be a nice activity in the elementary or middle school classroom.

May 28, 1998…

Hey, today in one of my Pre-algebra classes one of my students, Estefania Lopez, showed me a Trotter Date: her birthday, in fact, the day she was born! She used the “long” form of the date: March 13, 1984. So she wrote: 3/13/1984. Can you see why it is a Trotter Date Birthday (TDB)? Congratulations, Stefy!

Then later today another student, Andrew Kranstover, discovered that his date of birth was a TD as well. He was born on November 20, 1984. He also used the long form: 11/20/1984. Gee, isn’t it exciting to find two TDB’s in one day!!!

TEACHER’S GUIDE:

One suggestion to make this activity into a true “pre-algebra” item would be to present the process in the following manner:

Tell the students that here is a “formula” in which they will substitute certain values:

(m + d)x = y

where m = month number, d = day number, and y = year number. Then x, the solution of our equation, will be obtained by regular “algebraic” procedures. If it comes out as an integer, then we have a TD; otherwise not.

For example, let’s look at April 5, 1998. In the long form, we have 4/5/1998. So m = 4, d = 5, and y = 1998. By substituting, we have

(4 + 5)x = 1998
9x = 1998
x = 222

Wow! x is an integer! So, we have a TROTTER DATE.

The short form of the year (4/5/98) does not produce a TD. Observe:

(4 + 5)x = 98
9x = 98
x = 10.888…

In this way, we have raised the value of this activity to one that is getting ready for algebra, personalized (one’s birthday can be used), and deals with simple ideas in the regular math curriculum.