Math Problems Vault

Problem 1.14 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Additive Number Theory, Combinatorial Problems, and Applications"

Any weight that is a positive integral multiple of a given unit can be weighed with the weights $1, 2, 4, 8, 16, \ldots$ on one pan of the scales, and this can be done in exactly one way. That is, any positive integer admits a unique representation in the binary system.

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Problem 1.15 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Additive Number Theory, Combinatorial Problems, and Applications"

A set of weights $1, 3, 9, 27, 81, \ldots$ can be used to weigh any weight that is a positive integral multiple of a given unit if both pans of the scales are used, and this can be done in exactly one way.

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Problem 1.16 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Additive Number Theory, Combinatorial Problems, and Applications"

Write

(1+q\zeta )(1+q\zeta ^2)(1+q\zeta ^4)(1+q\zeta ^8)(1+q\zeta ^{16}) \cdots = a_0 + a_1 \zeta + a_2 \zeta ^2 + a_3 \zeta ^3 + \cdots .

Find the general formula for $a_n$ .

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Problem 1.17 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Additive Number Theory, Combinatorial Problems, and Applications"

Consider the expansion

(1-a)(1-b)(1-c)(1-d) \cdots = 1 - a - b + ab - c + ac + bc - abc - d + \cdots .

What is the sign of the $n$ -th term?

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Remark.

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E. Catalan, Problem: Nouv. Corresp. Math. Vol. 6, p. 143 (1880). Solved by E. Cesàro: Nouv. Corresp. Math. Vol. 6, p. 276 (1880).

Problem 1.18 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Additive Number Theory, Combinatorial Problems, and Applications"

Prove the identity

(1+\zeta +\zeta ^2+\zeta ^3+\cdots +\zeta ^9)(1+\zeta ^{10}+\zeta ^{20}+\cdots +\zeta ^{90}) \times (1+\zeta ^{100}+\zeta ^{200}+\cdots +\zeta ^{900}) \cdots = \frac{1}{1-\zeta } .

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Problem 1.184 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Various Propositions on Real Series and Sequences"

Every value $x$ of the interval $[-2,2]$ can be written in the form

x = \varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots }}}

where $\varepsilon_k$ , $k=0,1,2,\ldots$ , is either $-1$ or $+1$ . The representation is unique if $x$ is not of the form $2 \cos \dfrac {p}{2^q} \pi$ , $p, q$ integers, $0<p<2^q$ . These are the only numbers that may be written in the finite form

\varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots +\varepsilon _n \sqrt{2}}}} \, .

The finite representation can be extended to an infinite one in two ways: by putting either

\varepsilon _{n+1}=1, \quad \varepsilon _{n+2}=-1, \quad \varepsilon _{n+3}=\varepsilon _{n+4}=\cdots =1 ,

or

\varepsilon _{n+1}=-1, \quad \varepsilon _{n+2}=-1, \quad \varepsilon _{n+3}=\varepsilon _{n+4}=\cdots =1 .

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Remark.

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Cf.S. Pincherle: Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur. Vol. 53, pp. 745-763 (1917-1918); Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. 5, Vol. 27, 2nd Sem. pp. 177-183 (1918).

Problem 1.18.1 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 1.18 "[NO TITLE]"

The first and the third problem considered in the solution of 9 (concerned with $A_n$ and $C_n$ respectively) are the extreme cases of a common generalization which, properly extended, includes also 18. Formulate such a generalization.

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Problem 1.44 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 1.5 "Sequences of Real Numbers"

Let $p$ be a natural number greater than 1 , and $x$ a real number, $0 \leq x \leq 1$ . Show that there is a sequence $\left\{ a_{n}\right\}$ of integers with $0 \leq a_{n} < p$ for each $n$ such that

x = \sum _{n = 1}^{\infty } \frac{a_{n}}{p^{n}}

and that this sequence is unique except when $x$ is of the form $q / p^{n}, 0 < q < p^{n}$ , in which case there are exactly two such sequences. Show that, conversely, if $\left\{ a_{n}\right\}$ is any sequence of integers with $0 \leq a_{n} < p$ , the series

\sum _{n = 1}^{\infty } \frac{a_{n}}{p^{n}}

converges to a real number $x$ with $0 \leq x \leq 1$ . If $p = 10$ , this sequence is called the decimal expansion of $x$ . For $p = 2$ it is called the binary expansion; and for $p = 3$ , the ternary expansion.

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Positional number systems (binary, balanced ternary, decimal)

References