The Fermat Number Problem is a mathematical problem that asks you to find the largest Fermat number that can be expressed as the sum of two squares. Here is an example of how to solve this problem in Python:

```
def fermat_number(n):
# calculate the nth Fermat number
fermat = 2**(2**n) + 1
# iterate through all possible values of a and b
for a in range(1, int(fermat**0.5)):
for b in range(a, int(fermat**0.5)):
# check if a^2 + b^2 equals the Fermat number
if a**2 + b**2 == fermat:
return fermat, a, b
# if no solution is found, return None
return None
# find the largest Fermat number that can be expressed as the sum of two squares
for n in range(0, 10):
result = fermat_number(n)
if result:
print("Fermat number:", result[0])
print("a:", result[1])
print("b:", result[2])
break
```

In this solution, we first define a function `fermat_number`

that takes an integer `n`

representing the exponent of the Fermat number to calculate. The function calculates the nth Fermat number using the formula 2^(2^n) + 1.

The function then iterates through all possible pairs of integers `a`

and `b`

such that a <= b and a^2 + b^2 is less than the Fermat number. If a^2 + b^2 equals the Fermat number, the function returns a tuple containing the Fermat number, `a`

, and `b`

.

If no solution is found, the function returns `None`

.

In the main part of the script, we iterate through values of `n`

from 0 to 9 and call the `fermat_number`

function with each value. The first time a solution is found, we print out the Fermat number, `a`

, and `b`

, and break out of the loop.

To use this code, simply run the script. The output will be the largest Fermat number that can be expressed as the sum of two squares, along with the values of `a`

and `b`

that satisfy the equation a^2 + b^2 = Fermat number.

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