Five Related Prime Number Puzzles
(one has an infinite solution)

  

These first four puzzles require that all prime numbers be positive

• Find the smallest number of prime addends that add up to 1000 (any even sum > 7)
• Find the shortest list of different prime numbers that add up to 135 (any sum ending in 7 & other listed sums )
• Find the largest number of prime addends that add up to 1000 (any large sum)
• Find the largest number of different prime addends that add up to 1000 (any large sum)

This fifth puzzle allows both positive and negative prime numbers.

• What is the largest number of different prime addends that add up to 1000 (any sum)

And, the nice thing about this puzzle is that, after you've seen how I solved these puzzles for 1000, you can solve different puzzles for different target sums, like 125, 200, 500 or any number of your choice.

We will go through each puzzle, one at a time. Before you click here for my first solution, why don't you try to solve these puzzles (or at least solve the first one). At the bottom of each page, I'll repeat what puzzle will be solved next.