This page is a temporary display case of items that have no specific or permanent home. I hope it's not too confusing. I will add things to it whenever something catches my attention. So it's definitely **not** a final product.

If you want to go to Xmas Tree Primes, click here. |

If you want to go to OP-PO Primes, click here. |

The first item is a collection of interesting number facts. It should be self-explanatory. They were sent to me by Carlos Rivera (June 26, 2001). We had been discussing twin prime pairs whose sums were squares and cubes. Carlos kindly sent to me the smallest such pairs that produce powers beyond e = 3, that is, up to the 10^{th} power. He wrote:

Here are the earlier solutions for k^{e} = p+q, p & q twins:

6C Rivera^{2}= 17 + 19

2^{3}= 3 + 5

150^{4}= 253124999 + 253125001

96^{5}= 4076863487 + 4076863489

324^{6}= 578415690713087 + 578415690713089

6^{7}= 139967 + 139969

1518^{8}= 14097567309074239886172287 + 14097567309074239886172289

174^{9}= 73099303486215558911 + 73099303486215558913

168^{10}= 8954942912818222989311 + 8954942912818222989313

Here is some more trivia that was shared with me in June 2001 by two friends. We were discussing the infamous number of the beast: 666.

G. L. Honaker, Jr. observered:

There are exactly 6 6s in 666

To which I replied:

But here's more, from a JRM article by Charles Trigg, "A Closer Look at 37". [i've lost my copy, it was back in the early '70s.]Then Monte Zerger came back with these beautiful items:

The initials of the only U.S. president (so far, anyway) to resign while in office are R.M.N. He was the 37th president. Ok?

Now R = 18, right? And 18 x 37 = 666.

But wait--- there's more. M = 13 and N = 14, so M + N = 13 + 14 = 27. and 27 x 37 = 999 (which is... well, you know...).

Trigg's political message should be obvious, agreed?

In looking at 666

1. The first eight digits of 666

8 + 7 + 2 + 6 + 6 + 0 + 6 + 1 = 36

3 + 4 + 5 + 6 + 2 + 3 + 6 + 1 + 6 = 36

2. The six 6's total 36 as do the the other eleven digits.

6 + 6 + 6 + 6 + 6 + 6 = 36

8 + 7 + 2 + 0 + 1 + 3 + 4 + 5 + 2 + 3 + 1 = 36

3. The only digit missing in 666^6 is 9, the "upside down" version of 6.

Monte

To: 47society@yahoogroups.com

From: trottermath@gmail.com

Date: Fri Jul 6, 2001 1:32 am

Subject: Kaprekar & Counting Letters

Now isn't that an unusual subject line!

I assume that many readers are familiar with the unique number activity that leads to the famous result called Kaprekar's Constant: 6174. That is the one where you first order the digits of a four- place number from large-to-small, then reverse that number, and subtract the two. If you repeat the process correctly, within 7 steps you arrive at 6174 -- always. And there's 47 in reverse!

To find out more, do a search on Kaprekar, or better yet, just go to my webpage at:

It's all explained there.

But wait, there's more. Take Kaprekar's Constant and separate it into two parts... 61 and 74. Reverse the 61, yielding 16.

Next, write down all the composite numbers less than or equal to 16, i.e. 4, 6, 8, 9, 10, 12, 14, 15, & 16.

Now, express those in word form: four, six, eight, nine,...,sixteen.

Count the letters you have just written. If you were a good speller, you should have 47 (the reverse of 74)!

That's all there is to it.

Terry

Another pair of tib-bits from the Monte Files...

On February 8, 2000, he wrote:

I've encountered two things recently that may be of interest to you. They seem to "fit" with other things you have played with and made use of on your web page, etc.

- For what natural numbers is the arithmetic mean of divisors an integer? The smallest such number, other than 1, is 6 since (1+2+3+6)/4=3. The sequence begins 1, 6, 140, 270, 672, 1638, * As you can see the collection is relatively sparse.

- Consider all triangles with integral sides that can be formed having a given perimeter. Two triangles are considered different if the sides are permuted. So, for example, there is only one triangle with perimeter 3 and that's the (1,1,1) triangle. There are no triangles with perimeter 4. The triangles of perimeter 5 are (1,2,2), (2,1,2) and (2,2,1). The only triangle of perimeter 6 is (2,2,2). There are six triangles of perimeter 7 and they are (1,3,3), (3,1,3), (3,3,1), (2,2,3), (2,3,2), and (3,2,2). Amazingly (at least to me!) the number of such triangles is always a triangular number. For example, there are 21 triangles with perimeter 13 and 21=1+2+3+4+5+6.

I like the concept of using the nine digits 1 to 9 to make interesting number facts. (See The Century Problem.) Here is a nice variation that maybe should be called the "Century - 1" problem.

[From Prime Curios: 999]

A Xmas Tree Prime is hereby defined as "a prime number consisting of a 'triangular number' number of digits (*i.e.*, 1, 3, 6, 10, ...), such that the *first* digit is a prime, the following *two* digits (which constitute a prime by themselves) form a prime of 3 digits, then the following *three* digits (which constitute a prime by themselves) form a prime of 6 digits, and so on."

An example will clarify that definition. 241823 is a 6-digit Xmas Tree Prime because 2 is the single-digit prime, 41 is a two-digit prime, and 823 is a three-digit prime. But the first two primes also form a prime in their own right: 241.

Now to explain the idea of the Xmas Tree...

Notice that if we arrange the digits of our prime like so...

4 1

8 2 3

we have the beginning of a "Xmas Tree" of sorts, with each row being a prime number. Simple, maybe. But we like it. [One small requirement for the primes: a row cannot begin with a zero (0).]

Xmas Tree Primes (XTPs) come in various sizes and degrees of attractiveness, just like the real evergreen trees that are an essential part of the Xmas season. It's clear that there are only 4 XTPs of one row -- the four single-digit primes (2, 3, 5, and 7). So whatever size other XTPs have, they must begin with one of these four primes. [Let's call them the "species" of the primes/trees.]

There are only 29 XTPs of 2 rows, i.e. 3 digits. (241 is an example of a tree of species "2"; 389 for species "3", and so on.) They are distributed among the four single-digit primes as follows:

Species | Number |

2 | 6 |

3 | 14 |

5 | 4 |

7 | 5 |

Total | 29 |

Using any convenient list of primes less than 1000, they are not hard to find. WTM has such a list available HERE.

But finding XTPs of 3 rows provides the prime hunter with a larger challenge, though not one that is unsurmountable, given enough time and a longer prime list. of course. For example, WTM has found 23 XTPs of the form *211abc*, beginning with 211151 and ending with 211997. Since 211 is the smallest XTP of 2 rows, it follows that 211151 is the smallest XTP of 3 rows. The largest XTP however of 3 rows is this beauty: 797977!

As mentioned above, some Xmas trees are more attractive and better shaped than others. The same might be said for XTP's as well. Different characteristics can be described to show how interesting some of them can be. One way is to look for 6 distinct digits. Of the 78 possible XTP's beginning with "2", we have found 11 of them to be so composed, beginning with 241739 and ending with 283769.

On the flip side of that idea is to notice XTP's that have only 2 distinct digits, such as that largest one mentioned above, with only 2 digits being used. And one of our personal favorites is this: 211313. Notice how the appearance of its digits might be described... "one 2; two 3's; and three 1's".

Though we admit to have only begun our research into this intriguing little topic, we have found one XTP that may be one of a kind hard to find, if not unique. It is the smallest case of the species "3": 311137. Look at it in its "tree form".

1 1

1 3 7

Observe the 3 digits that form the "left side" (311) and the 3 digits that form the "right side" (317) -- they're both primes as well. What a shining, sparkling tree this one is!

Finally, to close this initial installment to the exploration of XTP's, we present the smallest possible 4-row, 10-digit tree. It is 2111511013, [recall that 2, 211, and 211151 are all primes, too]. But look at its tree form:

1 1

1 5 1

1 0 1 3

Aren't we lucky here? *2111* and *2113* are twin primes who form the left and right sides of our tree. And who is in the "central heart" position? The 5! The lone odd number that can't serve in the unit's place of any multi-digit prime! Was this a case of beginner's luck? What do you think?

1 1

1 5 1

1 0 1 3

1 0 8 6 7

The letter "**O**" refers to the **O**rdinal position of a given prime in the list of all primes. For example, the **8 ^{th}** prime is

We can now form a new number by concatenating the ordinal value with the prime itself, thusly, **819**. This is a number of the **OP** type. If such number is then a prime as well, it is an **OP prime!**

We can also reverse the placement of the ordinal value and the prime, putting the prime first and the ordinal second. This yields a number of the **PO** type. For our example, we would have **198**. As before, if such number is also a prime, it is a **PO prime**.

Unfortunately, 8 and 19 do not form primes in either manner. (They were used just to show the concept.) But never fear. There are many **OP primes** and **PO primes** that can be found. Here are some small samples of each:

Order | Prime | OP prime? | PO prime? |

2 | 3 | 23-yes | 32-no |

3 | 5 | 35-no | 53-yes |

4 | 7 | 47-yes | 74-no |

6 | 13 | 613-yes | 136-no |

9 | 23 | 923-no | 239-yes |

It should be obvious that primes of the OP type occur more frequently than those of the PO type. But when you do find an OP prime that also forms a PO prime, well, now you really have something. That shall be called an **OP-PO Prime**. The smallest such case is the **121 ^{st}** prime:

because 661121 is also a prime. In fact, 121661 is the 30^{th} **OP prime**, and 661121 is the 10^{th} **PO prime**. Nice, huh?!

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