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Valentin Albillo · 06-20-2018, 09:30 PM

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Hi Mr. Horn, pier4r, ijabbott and everyone else interested:

After waiting a considerable number of days for Warbucks to post anything on this thread either replying to Mr. Horn or substantiating his assertions, I'm posting here my final considerations on it all. If you're interested, read on (Warning: Long Read):

(06-15-2018 02:12 AM)Joe Horn Wrote:
(06-14-2018 03:07 PM)Valentin Albillo Wrote: You're barking up the wrong tree, Mr. Horn.

Are you implying that Warbucks cannot produce a 21-digit Ping-Pong Number whose cube is also a Ping-Pong Number?

Of course I'm "implying" that Warbucks cannot produce a 21-digit solution to Mr. Horn's Mini-challenge (MC henceforth). Furthermore, I'm explicitly stating here that not only can't he do that but also he doesn't even seem to understand the sources he used to make the assertions he posted.

I'll explain in detail but first the relevant timeline:

- Mr. Horn posted an interesting MC, "Ping-Pong Cubes" (PPC henceforth), which asks for an HP calculator program that finds the first ten PPC as defined by him.

- Within a few hours, Didier posted a program for the HP Prime, which Mr. Horn immediately commented enthusiastically ("Wow ! [...] Very cool !")

- A few hours later I saw the MC and posted a 5-line program for the HP-71B which finds all 10 small solutions in 0.49 sec. I naively expected some comment by Mr. Horn, as my code includes a novel technique to check non-iteratively whether a number is a PPN or not, and also because Mr. Horn hosted in the very distant past a column called "The Titan File" dedicated to the HP-71B so I thought he would appreciate my solution all the more. Alas, Mr. Horn didn't see it fit to comment anything and he didn't post even simply to acknowledge he had read my solution either.

- Nevertheless (and as this is not the first time Mr. Horn ignores my solutions to one of his challenges) I still was interested in the last part of his MC (finding the 11th PPC) and did some original research (i.e., entirely done by myself, without searching the net or using external references) in order to either find the coveted 11th PPC, or else prove that there's none, or at least improve the lower bounds for future searches.

- Thus, two weeks after posting my 71B solution, I posted a 100+ line message with tables, the formulas which generated them (in HP-71B BASIC), an explanation of the techniques used, comments and conclusions, the most important being that except for the 10 small ones already known there is no PPN less than 10^20 that generates a PPC when cubed, so any further search should necessarily begin with PPN candidates at least 21-digit long.

- Again, I naively thought that Mr. Horn would appreciate my original research and so would post some comment or at least would acknowledge this time that he had read it, but it was not to be, he completely ignored it. Again.

- Then someone identified as "Warbucks" posted about a dozen lines of what on examination turned out to be nothing but ramblings where he included such utter bluffs as "If anyone is interested I could do a proof [...]" and "If there is interest, for completeness, I can generate the 21 digit cube answer".

- Somewhat surprised at those bold statements, I analyzed what he said in his post and discovered that it's all a bunch of assorted lines directly copy-pasted from the Internet, taken from publicly available sources which have nothing to do with Mr. Horn's MC (or even with alternating cubes) and mixed with unsupported assertions a.k.a. bluffing. Yet Warbucks, who seems unaware of the irrelevancy of the sources he's parroting, had nevertheless the guts to offer "a proof" (of what !?) and a 21-digit solution right after copying whatever mumbo-jumbo he thought looked relevant.

- To my utter surprise, Mr. Horn then went on and while he didn't ever reply to either my 71B solution or my extensive original research, he did immediately reply to Warbucks post, telling him that "that" (i.e., posting a 21-digit solution) "would be greatly appreciated."

Facepalm ... Needless to say, Warbucks never replied to Mr. Horn's plea (of course!), nor did he post any "proof", nor did he "generate the 21 digit cube answer" he offered, despite being encouraged by Mr. Horn to do so. To help readers of this post to understand why, let's analyze Warbucks post in detail for evidence supporting what I've just said about it:

Warbucks Wrote:"So if you wanted to know the number of terms < 10^n, you would use:
9*(5^n-1)/4 - 9 n>=2
So if you wanted to now the number of terms<100 (10^2) n=2, there would be 45 terms. If you wanted to know the number of terms 10^n < #terms < 10^(n+1) we have:
9*5^n"

All these formulas and values apply only to PPNs, not PPCs or even just alternating cubes, and he took them verbatim either from internet sources (IMO, OEIS, PDF documents, specialized webs/books, etc.) or even adapted them from my own original research post where I gave them as user-defined functions for the HP-71B (shifted versions of my FNC4 and FNC3 respectively), so absolutely nothing new or original here, just mere googling and pasting to sound authoritative.

Warbucks Wrote:There is indeed a solution. The general case is 2^a(5^b)(j). There are three cases. If anyone is interested, I could do a proof [...]

This is sheer bluffing, he doesn't seem to understand or care what is being sought (Ping-Pong Cubes generated by cubing a Ping-Pong Number) or, in case he's just dealing with mere alternating cubes, he doesn't seem to understand either that his internet sources have nothing to do with such cubes. Matter of fact, he simply copy-pasted what is said in this PDF document:

45th International Mathematical Olympiad - Solutions

(or one of its several versions available in the net and even in some specialized books), which gives the solutions to six problems of the 45th International Mathematical Olympiad which took place at Athens, Greece in July 2004. At the end of this PDF document we find Problem 6, which reads:

"6. We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers n such that n has a multiple which is alternating."

so this Problem 6 deals with alternating integers alright, i.e., Ping-Pong Numbers, but has nothing do to with their cubes or alternating cubes or cubes at all, much less with finding PPNs which have PPN cubes as well. Nothing-to-do. At all.

It seems Warbucks didn't understand or care whether the info was relevant or not, he possibly just found the PDF document in the net after some googling and then proceeded to regurgitate and copy-paste lines from it. In particular, quoting once more what Warbucks posted:

Warbucks Wrote:There is indeed a solution. The general case is 2^a(5^b)(j). There are three cases. If anyone is interested, I could do a proof[...]

which was copy-pasted from the Official Solution in the aforementioned document, namely the lines in the document that read:

"For the general case n=2^a*5^b*k, where k is not a multiple of 5 or 2 and a = 1"

and also from the second solution by Joel Tay (in the document as well) which includes the Cases 1,2a,2b and 3 which are the "There are three cases" which Warbucks mentions. Then Warbucks continues with:

Warbucks Wrote:It was said that the next term in the sequence is 21 digits [...] Also if there is interest, for completeness, I can generate the 21 digit cube answer.

where the "It was said" obviously refers to my post (because nothing is said in the PDF document or anywhere else on the net about a 21-digit PPN or PPC) but actually I never said that there's a 21-digit solution, I said that 21-digit PPNs are the lower limit to search for the 11th PPC. Also, an "answer" to what ? Not certainly to what Mr. Horn is looking for, the 11th PPC.

In conclusion, it seems that Warbucks either has no idea of what Mr. Horn's MC is about or else he does not care, nor does he seem to be capable of producing any relevant proof (of what?) or even understand that the PDF document or source he's allegedly parroting has nothing to do with alternating cubes.

He seems to be utterly clueless and certainly can't produce a valid 11th solution in the form of a 21-digit PPN which remains PPN when cubed, which is the main interest of Mr. Horn, or even simply produce a non-PPC 21-digit alternating cube, which is way way easier. For instance, these are the eight 21-digit non-PPC alternating cubes, found by a simpler variant of my code in next to no time:

       105858789694589210761 = 4730521^3
       123274525814321496103 = 4976887^3
       307810347014567834983 = 6751927^3
       325874149438941050527 = 6881503^3
       361858767434383618729 = 7126009^3
       656181898125276107096 = 8689766^3
       905292149232581638541 = 9673781^3
       943014381450761214161 = 9806321^3

and just for show, these are the smallest 25-digit and 30-digit ones, found very quickly using this same simple variant (essentially just removing the PPN constraint for the base number):

      1270525474365816309892523 = 108308147^3
      101652769272709496147418783432 = 4667020818^3

Why Warbucks did something like this, parroting a document he googled out and extracting lines from it and then offering "proof" and a "solution" while seemingly understanding nothing of it, is beyond me. I can't fathom what he's trying to achieve with this kind of behavior.

To end on a realistic note and contingent on there being some interest, in a few days I'll post updated details extracted from my Original Research, establishing new, improved lower bounds and commenting on several heuristics I've thoroughly researched and the conclusions that can be extracted from them. Oh, and last-but-not-least a quite solvable PPN/PPC-based counter-MC for Mr. Horn and interested readers (including Warbucks).

P.S.: By the way, the final conclusion to Problem 6 of that 45th IMO is:

      "Concluding all, we have shown that a positive integer n is alternating if and only if it is not a multiple of 20.".

Great. Now someone please ask Warbucks what this has to do with Mr. Horns's MC or with alternating cubes in general, even if not PPC.

Regards.
V.
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