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Trotter Numbers
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Trotter Primes |
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Caveat lector
This is to alert you, dear reader, that what follows is to be considered as a work "in progress", and not a finished piece of writing. WTM is still researching the patterns and curious facts that can be found and described in this simple sequence of numbers. We make no pretense that what you will see here is of any major importance to anybody except ourselves. But it has pleased us, and that's why we are doing it.
We welcome your own observations. If we find them interesting, we may include them on this page, or ones to follow. Do check back from time to time, as this page will probably be changing as the days, weeks, and months roll by. With this gentle warning, we now begin.
Thanks.
If you want to return to Page 1, click here.
First Steps
Since any kind of decent research needs some data to look at and analyze, we began by computing the first 200 Trotter Numbers in a spreadsheet. Then we went through the list of numbers (TN's) one-by-one, finding which ones were indeed primes (TP's). Here is a summary of some of our initial investigation.
In the first 200 TN's, there were exactly 74 TP's. This alone should alert any member of the 47 Society that something significant is happening here. Although 74 is just the reverse of the main number 47, it still counts as a valid "sighting".
Actually, the 74th occurs when n = 193. [So at least the next 7 are composites.] This means that
No. of TP's in first 200 TN's 74 37
------------------------------- = ------ = ------ = 37%
200 200 100
The number of primes in the first 200 natural numbers is only 46, yielding a mere percentage of 23%. So TP's appear more frequently (in the first 200 possible instances anyway) than regular primes in the set of natural numbers. [Why is this true?]
Next, there are 15 TP's in the first 200 TN's that end with the digits "47". Now that is a lot more interesting than the last item above; here the digits 4 and 7 are present and side-by-side in their proper alignment. But remember: there are always two such TN's in every decade -- do you see why? This means that in the first 20 decades, there are 40 TN's ending in 47. Anyway we can state:
No. of TP-47-enders in first 200 TN's 15 3
------------------------------------------ = ---- = --- = 37.5%
No. of TN-47-enders in first 20 decades 40 8
For whatever it is worth, we find it intriguing that these two percent numbers are so close to one another. What do you think?
WTM has an "angel" upstairs assisting us by collecting data with a simple program, written in UBASIC. This divine friend has informed us that there are
735 TP's less than 100,000,000. A later angelic missive provided this additional information: "There are 2038 TPs less than a billion." So TP's are definitely out there; we "just gotta find them".
Second Steps
The next idea to attract my attention was the idea of consecutive TP's. As was noted on Page 1, the sequence of TN's begins with 6 consecutive prime numbers. This is significant because in the set of natural numbers, after 2 & 3, there can be no more consecutive numbers that are both prime. Whereas in our sequence, we are not constrained by the odd-even-odd-even character of the natural numbers. Since by definition, all TN's end in 7, naturally, we might have clusters of 2 or more TN's that are prime. The question is: where are they, and how many TP's are in any given cluster?
However, to make our discussion easier, we need to define some new terms. There is an old saying that goes: "Two's company; three's a crowd." (Here "company" refers to a "boy-girl couple" and "crowd" refers to an additional individual, referring to the eternal love triangle.) Therefore, we will define that when we find two consecutive primes in the TN sequence, they shall constitute a "prime couple". And any group of three, or more, consecutive primes shall be called a "prime crowd".
We are left now with one special situation: a TP where the preceding TN and succeeding TN are both composite. Such a prime shall be called a lonely prime, as it is "alone". (Its TN neighbors are not TPs.)
Well, now, let's take a look at those 74 TP's that exist among the first 200 TN's.
| Type | Number | Total |
| Lonely TPs | 23 | 23 |
| TP Couples | 7 | 14 |
| TP Crowd (3) | 4 | 12 |
| TP Crowd (4) | 1 | 4 |
| TP Crowd (5) | 3 | 15 |
| TP Crowd (6) | 1 | 6 |
| Totals | 39 | 74 |
Of course, it seems proper that there should be more lonely primes than any other type. But, doesn't it seem strange that there should be more crowds of 5 primes than there are of 4 primes? There are, in fact, 3 times as many.
Let's take a moment here and consider those lonely primes. Who is the "loneliest" TP contained in our data of the first 200 TN's? We might begin by observing how far apart it is from its other two nearest TP brethen. If we count the number of composite TN's [aka TC's] less than the TP, and the number of TC's greater than the TP, then add those two values, we would get some measure of its isolation from other TP's.
Upon examining our data, we find that T(86) = 77447 is probably the loneliest TP in this range. [Just for the record, it is the 40th TP in general, 9th TP that ends in 47, and the 12th lonely TP prime. And by the way, did you notice that this TP is composed of only 4's and 7's? Nice, huh?] But when we note that there are 4 composites less than it and 10 composites greater, its measure of loneliness is 14. The next possible candidate for "loneliness honors" has a measure of 12. So, 77447 is our loneliest TP thus far verified. Surely there are larger lonely TP's, but their identities are not presently known.
Are you ready for a little lonely TP trivia? Think about the following then.
Our "angel", mentioned above, sent down this information to us: "The largest TP less than one billion is T(9998) = 999600047."
Based on the information given above, we can at least state that this TP is the 2038th in general, but its position in the list of 47-enders, nor its position in the lonely prime list, was not revealed to us at this time. However, it is rather easy to prove that it is, in fact, lonely. We just need to calculate its two neighbors, T(9997) and T(9999), and see if those are TP's or TC's. Here is what we found:
- T(9997) = 999400097 = 17 x 593 x 99137
- T(9999) = 999800017 = 59 x 181 x 251 x 373
And that's all there is to it -- T(9998) is lonely.
But, wait. There is more here than meets the eye.
The sum of the digits of 999400097 is 47.
The last two digits of 999800017, the last TN less than one billion, is the prime "17", just as the sequence of TN's began - with 17.
The sums of the digits of the two prime factorizations share an interesting relationship.
Adding the digits of 17, 593, and 99137 results in 54, whereas the sum of the digits of 59, 181, 251, and 373 is 45 (the reverse).
Additional trivia
Let's take a moment to "stop and smell the roses", and see if we can find some interesting trivia among our first 200 TN's. Sometimes the nicest things are actually close to our home, thus we don't need to take long, expensive trips to seek diversion.
We might begin by identifying a specific subset of the TN's, which we choose to call the set of Trotter-Bond numbers. It is composed simply of TN's that end in "007" [you surely know about the famous James Bond, don't you?] First, they appear with precise regularity --- every 10th TN is a T-B number. After that, we examine to see which ones are also primes as well. It turns out that some are (4007), and some aren't (1007). So the question now becomes: of the possible 20 T-B numbers at our disposal so far, how many are prime, and how many are composite? (Can somebody send me the answer to this question?)
Feedback: (7/24/01)
Bill Dehority, a true and faithful fan of WTM for a long time, has investigated the matter of T-B numbers further. He has discovered that among the first 100 T-B numbers, exactly 29 of them are prime! [Note: 29 is a prime.] Bill also commented: "...this is ... a little larger than the number of primes less than 100."
Thanks a lot, Bill. Great work!
More Feedback: (7/25/01)
Not one to let any grass grow under his feet, his "mathematical feet anyway", Bill continued his search for T-B primes for another lap of 100 T-B numbers! He found 20 this time. We have a total of 49 so far confirmed. And 49 = 72.
But something unique happened this time. There is a string of four consecutive T-B primes; they occur when n = 1780, 1790, 1800, and 1810. The longest string of consecutive cases in the first set is only 3, when n = 20, 30, and 40. Interesting data.
Hey, Bill! Run another lap, will ya?
Did you perhaps notice the striking similarity between the first T-B prime, 4007, and the main prime (47) that started this whole odyssey? It almost makes one wonder about other instances of this case, namely, a 4 and a 7 separated by an even number of zeros. That is, which of these are prime? As the factorization labor intensifies here, WTM will give you the answer this time.
| T(n) | TN value | prime or factorization |
| T(20) | 4007 | prime |
| T(200) | 400007 | 19 × 37 × 569 |
| T(2000) | 40000007 | 167 × 239521 |
| T(20000) | 4000000007 | prime |
| T(200000) | 400000000007 | 37 × 71 × 373 × 408217 |
Recall that one of the categories of TN's is for those that end in 47. This is due, of course, to the fact that squares have only six digits that they can end with: 0, 1, 4, 9, 6, & 5. So, let's look at the TN's that end with 07, 17, 47, 97, 67, & 57. Our objective this time is to separate the digits in this way: the digits to the left of any of those pairs will compose one number and our pair another. If both parts form prime numbers, we will have found our special trivia item.
For example, T(14) = 1967. Separating as just described yields 19 and 67, both of which are prime. However, be careful: 1967 is not prime.
If we continue along this line of investigation, at least through the first 200 TN's, we find a somewhat startling result. Only those TN's that end with 67 have this prime-prime property. Out of six possibilities, only one works. [Note: 57 could never work as it is composite to start with.] This is not saying that if a TN ends with 67, that the left portion is automatically prime; merely that if the prime-prime property is manifest, then 67 is the right portion. There is quite a difference, as you'll see as you check your list, as we did.
By now, you should be getting the idea that there are many ways these numbers can be searched and categorized as having unique and interesting properties. So we will just close out this section with one piece of trivia that we like particularly well: T(102) = 104047.
First, take note of these facts: 104047 is the 42nd TP, the 10th TP that ends in "47", and the 14th of the lonely primes. But now if we split the number into two parts, namely 1040 and 47, we have the main prime "47" that is the heart and soul of this webpage project and another number of "primary" importance in the economic affairs of every American taxpayer -- the 1040 form!!
Some Theorems
Any decent number sequence should provide us with material to formulate some interesting theorems related to them. Here are a couple of examples for TN's.
Theorem #1: No Trotter Number is a multiple of 3.
Now it should be immediately obvious that no TN could be a multiple of 2 or 5, by the very definition of its construction (namely, all TN's end in 7). And it should be almost as obvious that every 7th TN is a multiple of 7. But it seems a mite curious to us that the other "small" prime, 3, is never a factor of any TN! After all, in the set of natural numbers, every third number is a multiple of 3, many of which (27, 57, 87, 117, etc.) even end with 7. So why is it impossible to find a TN that is divisible by 3? The best answer is "that's just a property of Trotter Numbers".
In spite of that (and lucky for us!), this theorem is fairly easy to prove. So easy in fact, that we leave it to you as an excercise. [You see, WTM is basically an educational website, too, that presents problems and challenges for you to think about. Okay?]
Theorem #2: No Trotter Number is a multiple of 11.
Here again, something of a mystery; why should no TN be a multiple of 11 either? As was said before, it's another of those strange properties of Trotter Numbers. And again, the proof, though only a little bit more difficult than for the one for 3, is fairly straightforward and easy. Another exercise for you.
We can continue with this "impossible theme" of other primes that are not divisors of any TN. Can you discover which primes are involved? What is the "Final Frontier" to this topic? We really don't know the complete answer; perhaps you can help us.
Generalized Trotter Numbers
Let's not "rest on our laurels" now, as it were, and think that this is the end of everything. There is lots more that can be said about "Trotter Numbers". You see, a good math topic should provide us with ample opportunities for expansion.
Recall the basic idea so far has been the attachment of the digit "7" at the end of a square number. A natural extension would be the investigation of using cubes with 7. Here is the formal definition:
The "cubic set of Trotter Numbers" is a subset of the natural numbers, or positive integers, defined by the following rule:
T3(n) = 10 * n3 + 7, where n = 1, 2, 3, …
The sequence begins: 17, 87, 277, 647, 1257, 2167, 3437, …
Whenever a given T3(n) [aka T3N] is prime, it shall be called a Trotter Prime of the cube-kind (denoted T3P).
This sequence can be found in Sloane's On-line Encyclopedia of Integer Sequences and has its own reference number: A061679.
So now, it's back to the drawing board, to investigate the properties, trivia, and possible theorems that might arise in this sequence.
Surely by now, the idea has entered your mind: What about 4th powers, 5th powers, and so on, with a 7 attached? What we have here is a sequence of sequences. Well, of course, WTM has thought about that already, but we have decided to leave the research into those matters to such persons with nimble fingers to do all the necessary typing, as the numbers get rather large rather quickly.
All the foregoing can be combined into one general formula:
Tk(n) = 10 * nk + 7, where for a given k > 1, n= 1, 2, 3, …
But let's carry this train of thought onto some other "tracks". We used just "7" in all our work so far. And that's good; 7 is a nice number. However, let's not forget about the three other digits that can serve in the unit's place of multi-digit primes: 1, 3, and 9. This prompts us to generalize the basic concept even further with this formula for T-sequences:
Tk(n) = 10 * nk + u, where for a given k > 1 and a given u = 1, 3, 7, or 9, n= 1, 2, 3, …
Now there's certainly a lot of territory there for research!
Before we take a much needed break from all this exploring, we want to present one last formula for a T-sequence that we have examined and found interesting. It is:
Tt(n) = 10 * t(n) + 7, where t(n) = the nth Triangular Number [ = n(n+1)/2].
To learn more about Triangular Numbers in WTM, go to The "3M" Game and Some Identities for the Triangular Numbers.