Today's number is 23456789
This number is somehow prime. In particular, it is the largest known prime number with consecutively increasing digits.
But what makes this prime even more special is that it trivially answers Gelfand's fourth question.
For context, Israel M. Gelfand asked 4 questions about if the leading (most significant) digits of the powers of numbers (i.e. 2^n, 3^n, . . . , 9^n) ever form all possible rows of digits or follow a particular pattern over time.
The problem investigates the behavior of the most significant decimal digits of these numbers as n increases (n = 1, 2, 3, . . . ). The four specific questions are:
Will the digit 9 ever occur in the leftmost position for 2^n? As it turns out, yes. The first such occurrence is at 2^{53}, namely 9,007,199,254,740,992
Will the row 23456789 ever appear for n > 1? That is, for some n, is is true that the leading digit of 2^n is 2, leading digit of 3^n is 3, . . . , and the leading digit of 9^n is 9? As it turns out, the only integer n > 0 where this holds is n = 1 (a link to the paper proving this is below)
Will a sequence of all the same digits ever occur? By the same paper linked below, this does not hold for any n.
Will the decimal expansion of an 8-digit prime ever occur? That is, if you concatenate the leading digits of these powers, is the result ever prime? This is, in my opinion, the most interesting of the 4. Unlike the other 3 questions that took years to prove, the case n = 1 solves it. That is, 23456789 is prime. As a quick final note, there are precisely 17,596 integers in the set of decimal expansions, of which 1,127 are prime numbers.
Overall, I consider it to be an interesting number. 2 of Gelfand's questions were "trivial" in some sense, and the fact that the fourth problem is solved by a prime with consecutively increasing digits is fascinating to me.
Proof paper: "A Simple Answer to Gelfand's Question" https://www.jstor.org/stable/10.4169/amer.math.monthly.122.03.234
