Problem 8.3.1. As a homework exercise, Tanya was asked to come up with 20 examples of the form $*+*=*$, where $*$ should be replaced with different natural numbers (i.e., a total of 60 different numbers should be used). Tanya loves prime numbers very much, so she decided to use as many of them as possible, while still ensuring the examples are correct. What is the maximum number of prime numbers Tanya can use?
# Answer: 41.

Solution. Note that in each example, instead of asterisks, three odd numbers cannot be used, i.e., at least one even number must be used. There is exactly one even prime number, which is 2. Therefore, among the 60 different numbers in the examples, at least 19 even composite numbers will be used, and there can be no more than 41 primes.

As is known, there are infinitely many prime numbers. We will show how examples can be constructed so that exactly 41 prime numbers are involved.

- $2+3=5$ - 3 prime numbers are used;
- $7+11=18$ - 2 prime numbers are used;
- $13+17=30$ - 2 prime numbers are used;
- $19+23=42$ - 2 prime numbers are used;
- in each subsequent example, two consecutive prime numbers are used in the left part.

All 60 numbers in these 20 examples will be different: all odd numbers are different primes, and even numbers, except for 2, are sums that increase when moving from the previous example to the next, and therefore cannot repeat.