MathDB

We write

\{a,b,c\}

for the set of three different positive integers

a, b

, and

c

. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of

\{a,b,c\}

. We can then calculate the sum of the elements of each subset. For example, for the set

\{4,7,42\}

we will find sums of

4, 7, 42,11, 46, 49

, and

53

for its seven subsets. Since

7, 11

, and

53

are prime, the set

\{4,7,42\}

has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors,

1

and themselves. In particular, the number

1

is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set

\{a,b,c\}

that has that number of subsets with prime sums, and explain why no other three-element set could have more.

Problem Statement