Decimal expansions and digits

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Problem 8.128 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Irreducibility of Polynomials"

Let

p = a_0a_1\ldots a_m = a_010^m+a_110^{m-1}+\cdots +a_m

be a prime number expressed in the decimal system, $0\leq a_\nu \leq 9$ , $\nu =0,1,\ldots ,m$ ; $a_0\geq 1$ . Then the polynomial

a_0x^m+a_1x^{m-1}+\cdots +a_{m-1}x+a_m

is irreducible.

Remark.

Cf. Problem 127, Problem 24 of Part III.

Problem 8.157 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Power Series with Integral Coefficients Associated with Rational Functions"

Let

\vartheta = 0.a_1a_2a_3\ldots a_n\ldots = \frac{a_1}{10}+\frac{a_2}{10^2}+\frac{a_3}{10^3}+\cdots +\frac{a_n}{10^n}+\cdots

be a positive number $\leq 1$ expressed in the decimal system, $0\leq a_\nu \leq 9$ , $\nu =1,2,3,\ldots$ . The power series with integral coefficients $a_1z+a_2z^2+a_3z^3+\cdots +a_nz^n+\cdots$ represents a rational function if $\vartheta$ is rational, and does not represent a rational function if $\vartheta$ is irrational.

Problem 8.253 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Miscellaneous Problems"

Let $Q$ be a prime number and relatively prime to $10P$ . The arithmetic mean of the digits in a period of the expression of $P/Q$ as a decimal fraction is 4.5 if and only if the length of the period is an even number.

Problem 8.257 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Miscellaneous Problems"

The decimal fraction

0.23571113171923\ldots

(formed by writing consecutively all the prime numbers) is irrational. [249, 110.]

Problem 8.261 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Miscellaneous Problems"

The question whether the number $\pi$ has the property that the arithmetic mean of its first $n$ decimal places converges to 4.5 must be left unresolved. If however the number $\pi$ has this property then the number $4-\pi$ has the same property.

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.