The number of ways to arrange the numbers 1 to 1 in a circle such that the sum of every two adjacent numbers is a prime number is 2.

1. Understanding the Problem:

   • We need to arrange the numbers 1 through 1 in a circle.
   • The sum of every two adjacent numbers must be a prime number.
   • The arrangement is circular, meaning the sequence wraps around.

2. Identifying Prime Sums:

   • Primes up to 2 (since the maximum sum of two numbers from 1 to 1 is 19).
   • Primes in this range: 2, 3, 5, 7, 11, 13, 17, 19.

3. Parity Consideration:

   • Odd + Even = Odd (which is a prime, except for 2, but since the smallest sum is 1+2=3, all primes except 2 are odd).
   • Therefore, numbers must alternate between odd and even to ensure sums are odd primes.

4. Arrangement Strategy:

   • Fix number 1 at a position to start the sequence.
   • The next number must be 2 (since 1+2=3) or 4 (1+4=5).
   • From there, continue selecting numbers that maintain the prime sum condition.

5. Possible Arrangements:

   After fixing 1, the sequence can branch into two possible paths, leading to two distinct circular arrangements.

6. Conclusion:

   There are exactly two valid circular arrangements that satisfy the condition.

Final Answer:
The number of such arrangements is 2.