Ponder This Challenge - December 2025 - Sums of a prime and an even number

This puzzle was suggested by Giorgos Kalogeropoulos - thanks!

Given a natural number 𝑛, take the first 𝑛 odd primes (3, 5, 7, 11, …) and the first 𝑛 positive even integers (2, 4, …, 2𝑛).

Compute all possible pairwise sums between these two sets (𝑛² sums in total, all odd).

We denote by 𝑓(𝑛) the number of these sums that are prime. For example, 𝑓(5) = 16.

Your goal: Find 𝑓(10⁸).

A bonus "*" will be given for finding 𝑓(10⁹).

Solution

• Click here to view the solution

  The numerical solutions are

  ƒ(1⨀^8) = 972989871151789

  ƒ(1⨀^9) = 871⨀51873756928⨀5

  This riddle was pretty straightforward, and the challenge was in writing an efficient implementation. For the primes, a standard sieve could be used to obtain them. For the sums, we received solutions describing them as convolutions and using Fourier transform techniques which is a nice way to consider this problem.

Solvers

• *Lazar Ilic (1/12/2025 2:51 PM IDT)
• *Jiri Spitalsky (1/12/2025 4:34 PM IDT)
• *Daniel Chong Jyh Tar (1/12/2025 4:39 PM IDT)
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