IBM Ponder This #251 · March 2019

Every real number can be approximated as a ratio of two integers. For example, pi is close to 22/7 but even closer to 3138/999; and there is an even better approximation with a denominator < 1000 (what is it?).

We define m=BAT(x,N) as the denominator, m, of a fraction k/m which is the best approximation for x with denominator < N.

Let N1 be the number of electrons in one coulomb of electric charge.

Let N2 be energy (measured in electron volts) released in the fusion of 1,000 deuterium atoms and 1,000 tritium atoms to form 1,000 He-4 atoms and 1,000 neutrons.

Find two non-square integers n1 and n2 such that b1=BAT(sqrt(n1),N1)=BAT(sqrt(n2),N1) and b2=BAT(sqrt(n1),N2)=BAT(sqrt(n2),N2) and both b1 and b2 are prime numbers.

Bonus question: for the smallest solution that also satisfy BAT(sqrt(n1),10000)=BAT(sqrt(n2),10000), what is the connection between BAT(sqrt(n1),10000) and IBM? Hint: see April 2011's challenge.

Solution

Best opened after a real attempt

To be added.