Consider the number
$0.23571379391713739171393971379371799173739113791379391173917133713717793$ ...
The number is formed by the ending digits of the prime numbers. Is it known whether this number is irrational ?
2026-04-05 22:07:20.1775426840
Number made from ending digits of primes
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Daniel Shiu has proved (in "Strings of congruent primes", JLMS 2000) that there are arbitrarily long strings of consecutive primes in any given residue class. In particular, the number described above has arbitrarily long strings 11111..., 33333..., 77777..., and 99999... in its decimal expansion. This is enough to show that it is irrational. The same proof shows that the analogous number for any base $b\ge3$ is irrational.