Challenge:

from secret import a,b,p,flag
from hashlib import sha512
from pwn import xor
from Crypto.Util.number import isPrime

assert p == a**51 + 51*b**51 and isPrime(p) and a > 0 and b > 0
print(hex(p)[2:])
print(xor(flag,sha512((str(a)+str(b)).encode()).digest()).hex())

# 1cec7c3ff93ca538e71f334e83d905eabd894729a1b515b89dc2392036bc7e5d59fad2c35dbb0a8903c8bb2e9cd5e4779a92d3f361eb1ce9fa2530c47881a8719763f828360138373ffa2ce627f8ccad08e9b5ead178c614f7899adc6a14fa33aa2216c463a04041e78cffa2c68963c6ff422a076bedd32236282eb3bd1b7ba870a3b1c8f639cd536cba329b10a6cf7b4ef78bd11052ff8a0d432fb6d3b221742aa1da6914876e94aca5abdaeef30acdfc90cbc621245ad288a634e8bdf4152ea8ed0062c872ace7b4011dc5743fa9c424413f4e3e8fa5c5513fd4a711141f2ef68c01177f78945db623ac6cc762a6813f11cc278a143ea657bf309e281ef59048a29f279c9ad8b37f221ac06242f577bba985a2aaec051d95391a9681f905472f4e7d1322da492639ee4a5ac776a476cece55f9dfb782c1ef869deed2226691d3157fbb6b131968ebfb1fe5bc1e44a158f1e2321c001355cc9cb3344f6b09b78d965a807cd60d58a9efbab8c6d4f75c8e5ac7c9cf0e5409b55bb2133129272685913be02166c6bffe3747ccd186b6c26fc9f09
# 43edcf6275293ce97d716f49875c4bdba37f6ab30f15a53f09b72bf3816edf6b92618fb56d569d911b2f6fe7a36d4a854022dddf671dc89b4800bc1605822aab72d3


Solve:


\[a^{51} + 51 \cdot b^{51} = p\]

now work mod p:

\[a^{51} + 51 \cdot b^{51} \equiv 0\] \[-51 \cdot b^{51}\equiv a^{51}\] \[\sqrt[51]{-51 \cdot b^{51}}\equiv \sqrt[51]{a^{51}}\] \[\sqrt[51]{-51} \cdot b \equiv a\] \[\text{let } x = \sqrt[51]{-51}\] \[b\cdot x + k\cdot p = a\] \[b \begin{bmatrix}x \\ 1 \end{bmatrix} + k \begin{bmatrix}p \\ 0 \end{bmatrix} = \begin{bmatrix}a \\ b \end{bmatrix}\] 


from gmpy2 import iroot
from pwn import xor
from hashlib import sha512
proof.all(False)

p = 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
x = int(GF(p)(-51).nth_root(51))

M = Matrix([
    [x, p],
    [1, 0],
]).T.LLL()
a, b = M[0]
assert p == a**51 + 51*b**51
flag = xor(bytes.fromhex("43edcf6275293ce97d716f49875c4bdba37f6ab30f15a53f09b72bf3816edf6b92618fb56d569d911b2f6fe7a36d4a854022dddf671dc89b4800bc1605822aab72d3"), sha512((str(a)+str(int(b))).encode()).digest())
print(flag)
# TCP1P{prime_prime_prime_prime_prime_prime_prime_prime_prime_prime}