Recent Results

Previous ] Home ] Up ] Next ]

      

Corollary 1. There is at least one nontrivial PPDI in every base b > 2, except possibly where b = 18k and k is not a perfect square.

 

Corollary 1 seems positively goofy. Is there really something special about bases that are multiples of 18?  Likely not, and the exceptions in the corollary have bothered me for years. Recently, I decided to revisit the distribution of PPDIs. My conjecture, of course, is that all bases above 2 contain nontrivial PPDIs. Even if my conjecture is correct, however, proving it may be difficult. The foregoing proofs all rely on two- and three-digit PPDIs, but PPDIs that might lead to arguments that could plug the ugly hole in Corollary 1 seem hard to come by. For example, Dik Winter of Centrum voor Wiskunde en Informatica in Amsterdam recently discovered that the smallest non-trivial PPDI in base-90 (i.e., base-18k, where k = 5) is {73}{62}{15}{62}{83}{18}{39}{47}. (Dik Winter has a brief discussion of “Armstrong numbers,” a.k.a. PPDIs, and two interesting tables of such numbers. These can be found here.) Of course, there might be an existence proof for the conjecture that does not actually identify the numbers that make the conjecture true, but it is not at all clear what sort of argument might be constructed. Alternatively, there may actually be bases without multi-digit PPDIs, but it is equally unclear how to approach proof of this conjecture.

Mike Keith and I have begun to eliminate some of the restrictions of Corollary 1. What follows has resulted from our collaboration:

 

Theorem 9. There is at least one nontrivial PPDI in every base b = 36k2, where = 1, 2, ... .

Proof. { 12k2-4}{ 24k2-2}{ 1 } is a PPDI in base b = 36k2. Thus, {8}{22}{1} is a base-36 PPDI, {40}{92}(1} is a base-144 PPDI, etc.

 

Theorem 10. There is at least one nontrivial PPDI in every base b = 90k+18, where = 0, 1, ... .

Proof. { 18k+4 }{ 36k+8 } is a PPDI in base b = 90k+18. Thus, {4}{8} is a base-18 PPDI, {22}{44} is a base-108 PPDI, etc.

 

Theorem 11. There is at least one nontrivial PPDI in every base b = 90k+72, where = 0, 1, ... .

Proof. { 18k+14 }{ 36k+29 } is a PPDI in base b = 90k+72. Thus, {14}{29} is a base-72 PPDI, {32}{65} is a base-162 PPDI, etc.

 

Now we are ready to improve upon Corollary 1. Using that result and those above, we can easily prove the following:

 

Corollary 2. There is at least one nontrivial PPDI in every base b > 2, except possibly where b = 18k and all the following are true: (1) k is neither a perfect square nor twice a perfect square and (2) neither k-1 nor k+1 is divisible by 5.

 

Notice that we could prove that all bases above 2 have nontrivial PPDIs if we could show that bases for which k, k+2, and k+3 are divisible by 5 have PPDIs. The k case seems especially difficult. We know that when k = 5 and b = 90, the smallest nontrivial PPDI is eight digits long. Even assuming an eight-digit PPDI could be the basis of a theorem, finding and proving an appropriate conjecture is likely to be difficult. In any case, Winter has shown that no base between 2 and 1000 lacks nontrivial PPDIs.

Finally, Mike was able to prove another theorem that does little to close the existence gap, but which is interesting for its own sake:

 

Theorem 11. Let t be the nth triangular number, n > 1. (The triangular numbers are those representing the number of objects needed to form a triangle like that formed by bowling pins. The first triangular number is 1, the second—reflecting a row of 1 object and a row of 2 objects—is 3, etc. In the case of bowling, i.e., tenpins, there are four rows of 1, 2, 3, and 4 pins. The 4th triangular number is 10.) If t is a square triangular number, let its square root be r. Then number {r}{r}{0} is a PPDI in base n.

Proof. {r}{r}{0} represents the value rn2 + rn = r (n2 + n). Since the formula for the nth triangular number t is (n2 + n)/2, however, and since, by assumption t = r2, we may write this as 2r3. But, of course, the sum of the cubes of the digits of {r}{r}{0} is also 2r3, so {r}{r}{0} is, by definition, a PPDI.

 

Unfortunately, square triangular numbers are rather sparse. The first two PPDIs whose existence is asserted by the theorem are 6608 and {35}{35}{0}49.

 

Previous Home Up Next

 
Send mail to Lionel Deimel with questions or comments about Lionel Deimel’s Farrago.
Copyright © 2000-2017 by Lionel Deimel. All rights reserved.