- Prove that a repunit with a non-prime number of digits will not be prime.
- MIT Prove that every positive integer has some multiple whose decimal representation involves all ten digits \(\{0,1,\dotsc,9\}.\)
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MathSE Prove that there exists a repunit divisible by \(2027\).
- Putnam 1989 A1 How many primes among the positive integers, written as usual in base 10, are alternating 1’s and 0’s, beginning and ending with 1?
- Nick’s Mathematical Puzzles What is the 1000th digit to the right of the radix (decimal) point in the decimal representation of the number \(\big(1+\sqrt{2}\big)^{3000}?\)
- Putnam 2014 B1 A base \(10\) over-expansion of a positive integer \(N\) is an expression of the form \[ N = d_k 10^k + d_{k-1} 10^{k-1} + \dotsb + d_0 10^0 \] with \(d_k \neq 0\) and \(d_i \in \{0,1,2,\dots,10\}\) for all \(i\). For instance, the integer \(N = 10\) has two base 10 over-expansions: \(10 = 10 \cdot 10^0\) and the usual base 10 expansion \(10 = 1 \cdot 10^1 + 0 \cdot 10^0\). Which positive integers have a unique base \(10\) over-expansion?
- Putnam 2017 B3 Suppose that \(f(x) = \sum_{i=0}^{\infty} c_ix^i\) is a power series for which each coefficient \(c_i\) is 0 or 1. Show that if \(f(2/3) =3/2,\) then \(f(1/2)\) must be irrational.
- Putnam 2020 A1 How many positive integers \(N\) satisfy all of the following three conditions: (1) \(N\) is divisible by 2020, (2) \(N\) has at most 2020 decimal digits, and (3) the decimal digits of \(N\) are a string of consecutive ones followed by a string of consecutive zeros.
- Putnam 2020 B1 (edited) For a positive integer \(n\), define \(d(n)\) to be the sum of the digits of \(n\) when written in binary. For example \(d(13) = 1+1+0+1=3.\) Determine the value of this number modulo \(2020.\) \[\sum_{k=1}^{2020} (-1)^{d(k)}k^3 \,. \]
- Putnam 2010 A4 (edited) Prove that for each positive integer \(n\), this number isn’t prime. \[10^{10^{10^n}}\!\!\! + 10^{10^n}\!\! + 10^n - 1\]
- Putnam and Beyond §1.5 The 2023-digit number \(99\dots 99\) is written on a blackboard. Each minute, a robot performs the following operation, making random decisions where necessary: a number on the blackboard is chosen and factored into two factors \(x\) and \(y\), the chosen number is erased, and the numbers \(x+2\) and \(y-2\) are written on the board. Is it possible that at some point all of the numbers on the blackboard are equal to \(9\)?
- Putnam 2007 A4 Find all polynomials \(f\) with real coefficients such that if \(n\) is a repunit, then so is \(f(n)\).
- Putnam 1998 B5 Let \(N\) be the repunit with 1998 decimal digits. Find the thousandth digit after the decimal point of \(\sqrt{N}\).
- Putnam 2002 B5 A palindrome in base \(b\) is a positive integer whose base \(b\) digits read the same backwards and forwards; for example, 2002 is a 4-digit palindrome in base 10. Note that 200 is not a palindrome in base 10, but it is the 3-digit palindrome 242 in base 9, and 404 in base 7. Prove that there is an integer which is a 3-digit palindrome in base \(b\) for at least 2002 different values of \(b.\)
- Putnam 1987 A2 The sequence of digits \[ 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots \] is obtained by writing the positive integers in order. If the \(10^n\)-th digit in this sequence occurs in the part of the sequence in which the \(m\)-digit numbers are placed, define \(f(n)\) to be \(m\). For example, \(f(2)=2\) because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, \(f(1987)\).
- Putnam 2023 B2 For each positive integer \(n,\) let \(k(n)\) be the number of ones in the binary representation of \(2023 \cdot n.\) What is the minimum value of \(k(n)?\)
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Goormaghtigh Conjecture Prove that \(31\) and \(8191\) are the only two positive integers that can be expressed as repunits in multiple bases.
- Feit–Thompson Conjecture Let \(R_n^{(m)}\) denote the repunit with \(n\) digits expressed in base \(m.\) Prove that for distinct primes \(p\) and \(q,\) \(R_p^{(q)}\) never divides \(R_q^{(p)}.\)