The Theorem Wu (General Case)
Prove that: if p is prime and p> 2 , for any integer
[Special case: For p=2, k = 1 (only)]
General case :
for all pj satisfying the theorem.
p = 9 = 3×3
p = 21= 3×7
p = 27 = 3×9 = 3x3x3
p =105 = 3x5x7
p =189 = 3x7x3x3
[Mr. William Wu proved the non-general case by using the Binomial Theorem and Legendre’s Theorem]
I envisage below to prove for all cases
by using the Advanced Algebra “Galois Finite Field Theory” :
where p prime and k >=1, the Fundamental Theorem of Galois Finite Field states that
1. The Galois Finite Field GF(p) is a multiplicative cyclic group;
2. GF(q) is the Galois Field extension of GF(p).
Step 1: Identity Equation:
Except the first term X^q and the last term Y^q (both with coefficient 1), all the middle terms with coefficients as :
are divisible by q,
Apply the Identity Equation:
Since GF(q) is a multiplicative cyclic group of order q, we get
Step 3: General case
Apply the Step 2 equation [*],
Proof: By The Fundamental Theorem of Arithmetic, any number (even and odd numbers) can be factorized as a product of primes.
Since The Theorem Wu is true for primes only,
=> it is false for even numbers
=> by the General case above, it is true for all odd numbers.
Corollary (Proof as your exercise!)
If m an even number,
Note: Wolfram Alpha verification
For p any prime or product of primes:
Galois Fundamental Finite Field Theorem
This is an excellent example of marriage of ancient Chinese Math, Middle-Age French Maths (Fermat & Galois) and modern Chinese (Singaporean) Maths — across 2,000 years!
The Theorem Wu is more general than the Ancient Chinese Remainder Theorem in 2 CE, which was further improved by 17 CE Fermat’s Little Theorem.
The Theorem Wu proves any number is/isn’t
1) a prime or
2) a product of primes > 2 (this property is neither found in Chinese Remainder Theorem nor Fermat’s).