Solution: We seek the largest integer $ N $ such that $ 20 < N < 40 $, and $ N $ is the sum of distinct prime numbers.

February 22, 2026

Optimal Prime Sum: Finding the Largest Integer $ N $ Between 20 and 40 Using Distinct Primes

When tasked with identifying the largest integer $ N $ such that $ 20 < N < 40 $, and $ N $ is expressible as the sum of distinct prime numbers, prime mathematicians and puzzle enthusiasts turn to the strategy of combining prime numbers efficiently. In this article, we explore the solution method, verify all candidates, and reveal how 37 emerges as the largest valid $ N $.

What Does It Mean for $ N $ to Be the Sum of Distinct Primes?

A sum of distinct primes means selecting one or more prime numbers from the set of primes greater than 2 (since 2 is the smallest and only even prime), ensuring no prime number is used more than once in any combination. Our goal: maximize $ N $ under 40, strictly greater than 20.

Step 1: Identify Prime Numbers Less Than 40

First, list all prime numbers below 40: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

Note: Since the sum must exceed 20 and be less than 40, and 2 is the smallest prime, using it helps reach higher totals efficiently.

Step 2: Strategy for Maximizing $ N $

To maximize $ N $, we should prioritize larger primes under 40, but always ensure they are distinct and their total lies between 20 and 40.

Because larger primes contribute more per piece, start with the largest primes less than 40 and work downward.

Step 3: Try Combinations Starting from the Top

We search for the largest $ N $ via targeted combinations.

Try: 37

Can 37 be written as a sum of distinct primes?

Try with 37 itself: $ 37 $ → valid! It is prime, so $ N = 37 $
But can we get higher? 37 is less than 40 and greater than 20 — but 38, 39, 40 are invalid (37 is the largest prime, 38, 39, 40 composite).
Is 37 the maximum? Not yet — let's verify if 38, 39, or 40 (though invalid) can be formed — they can’t, so 37 is a candidate.

But wait — can we exceed 37 using combinations?

Try $ 31 + 7 = 38 $ → valid primes, distinct: $ 31, 7 $ → sum = 38 Try $ 31 + 5 + 3 + 2 = 41 $ → too big Try $ 29 + 7 + 3 = 39 $ → valid Try $ 29 + 7 + 5 = 41 $ → too big Try $ 29 + 5 + 3 + 2 = 39 $ → valid Try $ 23 + 11 + 3 + 2 = 39 $ — also valid

Now try $ 31 + 5 + 3 + 2 = 41 $ — too large Try $ 29 + 7 + 3 = 39 $ — valid

Now try $ 37 + 2 = 39 $ — valid, but sum = 39 Try $ 37 + 3 = 40 $ — but 37 + 3 = 40, and 40 is allowed? Wait: Is 40 expressible as sum of distinct primes? 37 + 3 = 40 — yes! Both primes are distinct primes. So $ N = 40 $, but wait — the problem requires $ N < 40 $. So 40 is invalid.

So $ N < 40 $, strict. So maximum allowed $ N = 39 $.

Now confirm: Can $ N = 39 $ be written as sum of distinct primes?

Try:

$ 31 + 7 + 1 $? 1 not prime
$ 29 + 7 + 3 = 39 $ — valid Yes! $ 29, 7, 3 $ are distinct primes summing to 39.

So $ N = 39 $ is valid.

Is 39 < 40 and > 20? Yes.

Can we get 40? Only via $ 37 + 3 $, which is valid — but 40 is not less than 40 → invalid.

Try $ N = 38 $: $ 31 + 7 = 38 $ — valid but less than 39.

$ N = 37 $: prime itself — valid, but smaller.

Now confirm: Is 38 the maximum possible? Try $ 31 + 5 + 2 = 38 $ — valid, but still less than 39.

Try $ 29 + 7 + 3 = 39 $ — confirmed.

Try $ 23 + 13 + 3 = 39 $ — valid.

Try $ 19 + 17 + 3 = 39 $ — valid.

So $ N = 39 $ is achievable.

But wait — 37 + 2 = 39 — also valid $ 31 + 7 + 1 $ — invalid But $ 29 + 7 + 3 = 39 $ — best so far.

Wait: 37 + 2 = 39 — valid, and smaller primes allowed — so 39 is valid.

Now: Is 40 valid? $ 37 + 3 = 40 $ — distinct primes — valid, but 40 not less than 40 → excluded.

Thus, the largest valid $ N $ under 40 is at most 39.

Can we achieve 38? Yes, but less than 39.

Can we achieve 39? Yes.

But is 39 the maximum?

Wait — try $ 31 + 5 + 3 = 39 $ — yes $ 29 + 7 + 3 = 39 $ — yes $ 23 + 13 + 3 = 39 $ — yes $ 19 + 17 + 3 = 39 $ — yes

All valid.

Now, is 39 the largest possible?

Try $ 41 $? Too big. Try combinations adding to 40 but under 40: only 39 works.

Wait — $ 30 $ is not prime, but can we use $ 37 + 2 = 39 $? Yes — 37 and 2 are distinct primes — valid.

So 39 is expressible, and no larger number under 40 satisfies.

But wait — 37 + 2 = 39, and 37 is prime, 2 is prime, distinct — valid.

Hence, 39 is expressible as sum of distinct primes, and it’s the largest such integer between 20 and 40.

Final Verification: List all primes between 20 and 40

23, 29, 31, 37 — all primes

Avoid 41, 43 — too big

Now test:

$ 37 + 2 = 39 $ — valid
$ 31 + 7 = 38 $ — valid but less
$ 29 + 7 + 3 = 39 $ — best

Try $ 23 + 13 + 3 = 39 $ — valid Try $ 19 + 17 + 3 = 39 $ — valid

No combination exceeds 39 within bounds.

Conclusion

The largest integer $ N $ such that $ 20 < N < 40 $ and $ N $ is the sum of distinct prime numbers is:

> 39

This is achieved via combinations such as $ 37 + 2 $, $ 29 + 7 + 3 $, and others.

Using the greedy approach from the largest primes downward confirms that 39 is the maximum possible.

Bonus: Why Not 40?

虽然 $ 40 = 37 + 3 $ is a valid sum of distinct primes, the condition $ N < 40 $ excludes it. Hence, 39 is the largest strictly smaller valid sum.

Key takeaway: To find the largest sum of distinct primes in an interval, prioritize largest primes, ensure distinctness, and verify bounds—39 is optimal.

Keywords: largest integer N between 20 and 40, distinct prime sum, largest prime sum under 40, solution prime sum 37+2=39, prime numbers sum 20<N<40

Meta description: The maximum sum of distinct primes between 20 and 40 is 39, achieved via combinations like 29+7+3 or 37+2. Learn how to identify such optimal values efficiently.