September 2 There's always something going on with prime numbers. Always. A few months ago I ran into a breed I had never met: digitally delicate primes. The name itself was charming, if also a mouthful. And then there are widely digitally delicate primes. How could I not be captivated?
So last Friday's (Aug 27) math column in Mint is about these creatures. I hope you meet them someday. Send me a postcard. Digitally delicately put: I don’t know, https://www.livemint.com/opinion/columns/digitally-delicately-put-i-don-t-know-11630008211205.html cheers, dilip PS: "Composite" doesn't quite mean the same as "non-prime", but I'll let that be for now. --- Digitally delicately put: I do not know The mathematician Murray Klamkin (1921-2004) taught at several universities, helped to organize various competitions ("Mathematical Olympiads") and won awards for his work. In 1988, there was the Mathematical Association of America's Award for Distinguished Service to Mathematics. Four years later, his years of contributions to Olympiads and the like won him the David Hilbert Award. But for all that, Klamkin was probably best known for proposing serious mathematical questions. Of course, it wasn't that he necessarily knew the answers, or at least not immediately. In offering these challenges, Klamkin's purpose was to open up new areas of research. Many of his questions made their way into books. Like one he co-authored with two other mathematicians, "Five Hundred Mathematical Challenges"; or like “Index to Mathematical Problems 1975-1979: A Compendium of over 5,000 Problems” by Stanley Rabinowitz and Mark Bowron; or like "Problems from Murray Klamkin", which two mathematicians published after his death. In January 1978, Klamkin sent Mathematics Magazine this question: "Does there exist any prime number such that if any digit is changed to any other digit, the resulting number is always composite?" (Composite means non-prime). For example, take 79, which is a prime number. I could change the 7 to 3, 4, 6, or 9 and get composite numbers every time (none of 39, 49, 69 and 99 are prime). But if I changed it to 1, 2, 5 or 8, we have another prime (19, 29, 59, 89). Similar reasoning applies to the 9 in 79. That is, 79 doesn't answer Klamkin's poser. Try 103, also a prime. Change the 1 to 5 and we have 503, which is prime; Klamkin is foiled again. He is similarly foiled by the primes 919, 4021 and 48281 - and, I'm willing to bet, any prime you name. So you can see why he asked the question. Aside: maybe you wonder why Klamkin or any other mathematician would want to ask and answer such a question. Of what possible use is it to the real world? I do not know, but I do know that it is just the kind of puzzle that gets mathematical juices flowing. End of aside. Anyway, Mathematics Magazine published Klamkin’s poser in their May 1979 issue. Below it was a solution from the remarkable Hungarian mathematician Paul Erdös, which began with words that probably wouldn't have surprised anyone who knew the man: “We prove a slightly stronger result.” Erdös’s proof calls on Linnik’s theorem and the “well-known theorem of Bang-Birkhoff-Vandiver” and plenty more, which is to say that by about its 3rd paragraph it has surpassed my mathematical capabilities. After a page-and-a-half of mathematical manipulations, Erdös writes “This completes our proof.” But he doesn’t stop there. In fact, he had telegraphed this at the start: “In the end [we will] make some comments and state a few more problems.” (Note how one problem stimulates more problems, something else that delights mathematicians). After his proof is done, Erdös writes these lines: “We can deduce by the methods used here that there are infinitely many primes [of a certain type P] so that if we simultaneously alter [some of their] digits we always get a composite number. Is it true that if [we take numbers of a certain type A] then there are always primes, in fact infinitely many of them, so that [numbers formed by adding primes P and numbers A] are composite?” Take a moment to digest that. In effect, Erdös has taken the puzzle into the realm of the infinite - there’s not just one prime that answers Klamkin, but an infinity of them - and that gets him thinking in other related directions. So he leaves an open question, also involving infinity, for other mathematicians to pursue. Over forty years later, two mathematicians at the University of South Carolina, Michael Filaseta and Jacob Juillerat, published a paper addressing this open question (“Consecutive primes which are widely digitally delicate”, arXiv, https://arxiv.org/pdf/2101.08898.pdf). Filaseta and Juillerat start by acknowledging Klamkin and Erdös. They remind readers that Erdös’s “infinitely many primes” that answer Klamkin’s question were, in 2016, given the delicious label “digitally delicate” by two other mathematicians. Then they write: “The first digitally delicate prime is 294001,” and you can check this for yourself. Change any of the digits in 294001 to another, and you get a composite number: for example, 294501 and 694001 are both composite. They also note that there are composite numbers that, when you manipulate them the same way, produce only composites. 212159 is the smallest such composite, and again, you can check this for yourself: for example, 216159 and 212179 are composites. But sticking with the digitally delicate: in 2020, Filaseta and Jeremiah Southwick proved something startling. You wouldn't typically do this, but you could write any given number with any number of leading zeros. For example, you could write the prime 79 as 079, 0079 or even 000000000079. All those are still 79, still prime; and in fact, you could prefix a number with an infinite number of leading zeros, and it would remain the same. Filaseta and Southwick showed that there are primes which answer Klamkin's question even if you write them this way, with any number of leading zeros. That is, "if any digit of [such a prime], including any one of [its] infinitely many leading zeros, is replaced by any other digit, then the resulting number is composite." If you think about it, this is nothing short of astonishing. Prefix one of these primes with any number of zeros, point to any one of them and change it to any other digit, and presto, like magic, what you get is composite. Filaseta and Southwick called such primes "widely digitally delicate." So is 294001, the first digitally delicate prime, also widely digitally delicate? No, because you could write it as 00294001, change that first 0 to 1, and 10294001 is prime. In fact, these widely digitally delicate primes are somewhat like black holes used to be - their existence is a theoretically proved certainty, but "no specific examples of widely digitally delicate primes are known." If that much isn't intriguing enough, what Filaseta and Juillerat proved in this paper should fit the bill. Never mind that we don't yet know of a single widely digitally delicate prime, let's just keep in mind that one exists. But it's not just that there's one. Nor is it that there are some such primes. Nor even is it that there is an infinite number of such primes, which there is. What these two mathematicians proved is that there are strings of consecutive primes - like 41, 43, 47 and 53 are four consecutive primes - that are all widely digitally delicate. But not just that, either. They showed that these strings can be of any length at all. That is, pick any number - 4, 89, 271,828, 31,415,926 - yes, any number at all, and there is a string of that many consecutive primes that are all widely digitally delicate. And we don't know any of them. Not one. That, by the way, is hardly a lament. I say so not only because it leaves mathematicians plenty to search for, not only because it gives us yet another glimpse of the meaning of infinity. Even more than those things, this appreciation of the unknown is what makes mathematics, and science more generally, so fascinating. I didn't tell you the four words at the end of Paul Erdös's proof above, after the question he asks. Those words are: "I do not know." -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. To view this discussion on the web, visit https://groups.google.com/d/msgid/dilips-essays/CAEiMe8rvPmzcMq1g-Gab8pi3BsqTB5fj4M6Yai35%2BJ%3D581gfLQ%40mail.gmail.com.