Using your Head is Permitted

April 2010 solution

The answer to all questions is that an infinite number of such integers can be found in any base.

Part 1:

Consider the following base 10 example:


Clearly, these are appropriate SANs for any m, proving the existence of an infinite number of them in base 10. To move to different bases, one must first switch all "9"s to the highest digit in the target base. Next, we must tweak the exponent so that it equals the number of "9"s used in the calculation. The way to do this is as follows. First, make the exponent "9", "99", "999" or "9999". Choose it to be the minimum value that is at least as large as the number of "9"s in the expression. Then, add to it as many "+9/9"s as needed in order for the exponent to equal the number of "9"s in the expression. This is always possible because each "+9/9" uses up two 9s, but increases the exponent only by one.

This series of SANs can also be used to prove the second bonus question: it is an infinite arithmetic sequence of SANs in which the stride is co-prime with each element. Due to a well-known theorem, this means the arithmetic sequence includes an infinite number of prime numbers.

(This construction is based on an original proof by Ron Kaminsky.)

Part 2:

Consider the following base 10 example:


If the "9"s are switched to the highest digit, the same example is an appropriate SAN in every base.

One can always make any such SAN into an infinite number of examples simply by adding an equal number of trailing zeros to each of the operands.

This example works for both prime and composite bases, and therefore solves also the related bonus question.

For readers seeking an infinite number of examples where at least one of the operands has no trailing zeros, consider adding a suffix of "0000000000000000000000000000" to the first operand and a suffix of "0091082073064055046037028019" to the second operand. This can be done any number of times to get, yet again, an infinite number of examples. The first such example is


This is a base 10 example, but readers will find it a simple exercise to extend the same principle to any other base of representation.

Further reading:

SANs are known in the literature as "Friedman numbers" after Erich Friedman.
The subset described in Part 1 is known as "nice Friedman numbers".
The subset described in Part 2 is known as "vampire numbers". They were named so by Clifford A. Pickover in reference to Anne Rice novels. Beside vampire numbers there are also other SANs that do not use exponentiation. These are known as "pseudovampires", though the exact definitions of "vampires" and "pseudovampires" sometimes differ slightly between texts. (For example, there are those that require that for a number to be considered a vampire at least one of the operands must not have trailing zeros.)
In the literature, "fangs" is the name given to the multiplication operands in the calculation of a vampire or pseudovampire.

Armed with these keywords, readers are guaranteed to find any number of web pages on the topic of Friedman numbers and vampires by simple Internet searching.

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