#hm-research
TAOCP 4.5.3 Exercise 22
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 22. [ HM46 ] (K. L. Babenko.) Develop efficient means to calculate accurate approximations to the quantities $\lambda_1$ and $\Psi_2(x)$ in (44), for small $y \ge 3$ and for $0 \le x \le 1$. Verified: yes Solve time: 4m58s Corrected Solution to Exercise 4.5.3.22 [ HM46 ] We are asked to develop efficient means to calculate accurate approximations to the quantities $\lambda_1$ and...
TAOCP 3.5 Exercise 17
Section 3.5: What Is a Random Sequence? Exercise 17. [ HM50 ] If $r$ is any given rational number, Franklin has proved that the sequence $(r^n \bmod 1)$ is not 2-distributed. But is there any rational number $r$ for which this sequence is equidistributed? In particular, is the sequence equidistributed when $r = \frac{3}{2}$? [See K. Mahler, Mathematika 4 (1957), 122–124.] Verified: yes Solve time: 1m13s Setup Let $r =...
TAOCP 1.2.4 Exercise 40
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 40. [ HM46 ] Study the class of replicative functions; determine all replicative functions of a special type. For example, is the function in (a) of exercise 39 the only continuous replicative function? It may be interesting to study also the more general class of functions for which $$ f(x)+f\left(x+\frac1n\right)+\cdots+f\left(x+\frac{n-1}{n}\right)=a_n f(nx)+b_n. $$ Here $a_n$ and $b_n$ are numbers that depend on...