Posted by: Gary Ernest Davis on: April 17, 2011

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The

It shares this property with since .

What makes even more curious is that it is the only number other than that has this property.

To prove this we need to talk a little more exactly about what is this property.

The Münchausen property

Suppose is a positive integer and $latex d_k, see ldots d_1, patient d_0$ are the base 10 digits of n.

This means – a fact every elementary teacher of mathematics tries to get across to students when teaching place value – that

So given that (base 10) we form a new number:

The property we are interested in – sometimes called the Münchausen property – is that .

So we see that .

We will now see that there is no other positive integer for which .

But first, store a little excursion into properties of the function .

We should have discussed what happens if a digit of the positive integer is : because, then, we have to assign a value to .

There is no magic way to decide how to assign a value to – we will use the continuity of the function for :

Because the limit of as approaches with is we assign the value .

There is nothing magic about this – it just turns out convenient for our purposes here to have .

We can see the initially decreases as increases, reaches a minimum value, and then increases.

We can see this from the derivative of , which we can calculate by first taking (natural) logarithms:

so and therefore

This is negative for and positive for .

So, for non-negative integer values of the function is non-decreasing: in fact, apart from , we have for all positive integers .

For a positive integer the number of (base 10) digits of is the floor of , the largest integer less than, or equal to, .

This means that the number of digits of is less than .

If the base 10 digits of the positive integer are then:

- for
- The number of digits of is

So,

This says that cannot grow too big in terms of the number of digits of .

We look at the behavior of the function :

The derivative of is and this is positive for .

So, for we have:

x

In other words, for we have

x

If is too big – for instance, – then cannot be equal to .

The reason is that for we have while

So we search through all positive integers up to and find no integers with other than .

Suppose instead of forming as we did, we move the digits forward, with the first digit moving around to be last: in other words, we form a new number .

Do we ever have ?

Well, yes: we do for because and

What if we move each digit forward two (with the first two cycling around to be be last and second last)?

van Berkel, D. (2009) On a curious property of 3435. Retrieved from arxiv.org: On_a_curious_property_of_3435 [This article provides the argument I have described in this post]

Perfect digit-to-digit invariant Wikipedia.org [In this reference another number is counted as a Munchausen number due to their using the convention 0^0=0]

This is so amazing. I wonder what other properties 3435 has?

1 | no name

April 23, 2011 at 10:13 pm

You link to Wikipedia, which gives 438579088 as another Munchausen number. Did you even read it?

Gary Ernest Davis

April 23, 2011 at 10:50 pm

Yes, I did read it.

You will see that there they choose to define 0^0=0, which is how this other number comes to be counted as a Munchausen number.

I explained how for the proof in the post I was taking 0^0=1 and why I was doing that.