Observations & Theorems

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It is obvious from the table of PPDIs that PPDIs are not exactly rare. The number of PPDIs seems generally to increase with base, and the order of the largest PPDI seems generally to increase with base. Specifically, it appears that (1) all bases contain PPDIs, and (2) it may be possible to bound tightly the highest order PPDI as a function of base. Observation (1) is treated below. Those interested in examining observation (2) should study the article in Journal of Recreational Mathematics referred to elsewhere.

HPDIs are not rare either, there being between 0 and 5 in bases 2-10.



It is possible to prove some properties about the distribution of PPDIs. In what follows, the notation { f } refers to the numeral whose value is f. (The braces may be omitted where no ambiguity results.) In what follows, we do not necessarily distinguish between a number (i.e., an abstract value) and its representation in a given base. In all cases, the meaning should be clear.


Theorem 1. Every odd base (3, 5, ... ) contains at least one nontrivial (multi-digit) PPDI.

Proof. It is easy to show that { k+1 }{ k+1 }2k+1, where k = 1, 2, ... is a PPDI. In particular, in base-(2k+1), { k+1 }{ k+1 } represents the value (k+1)(2k+1) + (k+1) or 2k2 + 4k + 2. But this is just (k+1)2 + (k+1)2, which, by definition, makes k+1 }{ k+1 } a PPDI in base-(2k+1). Thus, 22 is a base-3 PPDI, 33 is a base-5 PPDI, etc.



In general, such existence theorems are proved by showing that the positional value and summation value of numbers in a particular form are equal. The details are tedious and are omitted in the proofs that follow.


Theorem 2. There is at least one nontrivial PPDI in every base b = 3k+1, where k = 1, 2, ... .

Proof.k }{ 2k+1 }0b is a PPDI. Thus 130 is a base-4 PPDI, 250 is a base-7 PPDI, etc.



Theorem 3. There is at least one nontrivial PPDI in every base b = 3k+2, where k = 1, 2, ... .

Proof.k }{ 0 }{ 2k+1 }b is a PPDI. Thus, 103 is a base-5 PPDI, 205 is a base-8 PPDI, etc.



Theorem 4. There is at least one nontrivial PPDI in every base b = 18k+6, where k = 0, 1, ... .

Proof. { 14k+5 }{ 4k+1 }{ 12k+4 }b is a PPDI. Thus, 514 is a base-6 PPDI, {19}{5}{16} is a base-24 PPDI, etc.



Theorem 5. There is at least one nontrivial PPDI in every base b = 18k+12, where = 0, 1, ... .

Proof.10k+6 }{ 8k+6 }{ 12k+8 }b is a PPDI. Thus, 668 is a base-12 PPDI, {16}{14}{20} is a base-30 PPDI, etc.



Theorem 6. There is at least one nontrivial PPDI in every base b = 18k2, where = 1, 2, ... .

Proof. { 9k2-3k }{ 9k2 }b is a PPDI. Thus, 69 is a base-18 PPDI, {30}{36} is a base-72 PPDI, etc.



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.

Proof. The result follows directly from the preceding theorems.



The above corollary is hardly satisfactory. No evidence suggests the existence of bases without PPDIs, but Michael Jones and I were unable to remove the restrictions of the corollary. I hope eventually to prove that there is at least one nontrivial PPDI in all bases b > 2. Anyone who thinks he or she can prove (or disprove) this conjecture should contact me immediately.

Here is another interesting result.


Theorem 7. There are infinitely many numbers that are nontrivial PPDIs in at least two bases.

Proof.k }{ 2k+1 }{ 0 }3k+1 = { k }{ 0 }{ 2k+1 }3k+2, where k = 1, 2, ... . This follows directly from theorems 2 and 3. For example, 1304 and 1035 are both PPDIs, and 28 = 1304 = 1035. Of course, no fixed string of numerals—except length-1 strings—can be a PPDI in two different bases, as the sum of the powers of the digits is characteristic of the numeral string itself.



Returning to an issue raised in Distributions, we can show that, at least for some bases, the number of PDIs is infinite.


Theorem 8. There are infinitely many bases in which there are infinitely many PDIs.

Proof. It is easy to verify that the (j+1)-digit representation { i }{ 0 } ... { 0 } (that is, the numeral representing i followed by j zeroes) in base i2 is a PDI of order 2j+1, where i > 1 and j > 0. It follows that 2 is a base-4 order-1 PDI, 20 is a base-4 order-3 PDI, 200 is a base-4 order-5 PDI, etc. Likewise, 3 is a base-9 order 1 PDI, 30 is a base-9 order-3 PDI, 300 is a base-9 order-5 PDI, etc. In fact, i2 in the proof can be replaced by i3, i4, etc., to generalize a bit more. Theorem 8 makes its point as written, however.



Finally, Michael and I were asked if there are any prime PPDIs in base-10. It isn't clear to my why this should be an interesting question—primes are easily identified in smaller bases in the table of PPDIs—but, for those who are interested, I note that the 14-digit PPDI 28,116,440,335,967 is prime.


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