## It is possible to write ten as the sum of primes in exactly five different ways:

\begin{align}

&7 + 3\\

&5 + 5\\

&5 + 3 + 2\\

&3 + 3 + 2 + 2\\

&2 + 2 + 2 + 2 + 2

\end{align}

What is the first value which can be written as the sum of primes in over five thousand different ways?

### To solve this problem, we can use dynamic programming. We will create a list where the value at each index represents the number of ways to express that index as the sum of primes.

Here’s the Python code to find the first value which can be written as the sum of primes in over five thousand different ways:

“`python

import math

def count_primes(n):

primes = [True] * (n + 1)

primes[0] = primes[1] = False

for i in range(2, int(math.sqrt(n)) + 1):

if primes[i]:

for j in range(i * i, n + 1, i):

primes[j] = False

return primes

# Function to count the number of ways to express a number as the sum of primes

def count_ways(n, primes):

dp = [0] * (n + 1)

dp[0] = 1

for i in range(2, n + 1):

if primes[i]:

for j in range(i, n + 1):

dp[j] += dp[j – i]

return dp

def find_value():

primes = count_primes(100) # Assuming that the first value will be within 100

n = 2

while True:

ways = count_ways(n, primes)

if ways[n] > 5000:

return n

n += 1

# Find the first value which can be written as the sum of primes in over five thousand different ways

result = find_value()

print(result)

“`

This code first checks for prime numbers up to a given limit (100 in this case) using the count_primes function. It then uses the count_ways function to calculate the number of ways to express each number as the sum of primes, using dynamic programming.

The find_value function starts from 2 and increases the value of n until it finds a number that can be written as the sum of primes in over five thousand different ways. The result is then printed.

Note: The limit 100 used in count_primes can be adjusted based on your requirements. If you need to find the value which can be written as the sum of primes in over a larger number of ways, you may need to increase this limit.

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