Odd Primes Congruent to 1 or 3 mod 4: Proof

  • Context: Undergrad 
  • Thread starter Thread starter Lenardo
  • Start date Start date
  • Tags Tags
    Prime
Click For Summary
SUMMARY

All odd prime numbers are congruent to either 1 or 3 modulo 4. This conclusion is supported by the Division Algorithm, which illustrates that odd integers cannot be congruent to 0 or 2 modulo 4, as those values are reserved for even integers. The discussion confirms that the statement holds true for all odd integers, not just primes, simplifying the proof to the properties of odd numbers.

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with the Division Algorithm
  • Basic knowledge of prime numbers
  • Concept of odd and even integers
NEXT STEPS
  • Study the Division Algorithm in detail
  • Explore properties of modular arithmetic
  • Research proofs related to prime number congruences
  • Learn about the distribution of odd and even integers in number theory
USEFUL FOR

Mathematicians, students studying number theory, and anyone interested in the properties of prime numbers and modular arithmetic.

Lenardo
Messages
3
Reaction score
0
Is the following statement true ?

Any odd prime number is congruent to either 1 or 3 mod 4.

If yes , then how we could prove it ?
 
Physics news on Phys.org
It's not as hard as it looks. Think of some actual numbers that are congruent to 2 or 0 modulo 4.
 
Aha.. I read before couple of minutes something called "Division Algorithm" ..and I can see why ? thank you
 
Are you still trying to prove that odd primes = 1 or 3 mod 4? I'm glad if you solved that now.
 
You can simplify it to "all odd numbers are congruent to 1 mod 2." To answer your question: yes, all odd primes adhere to the rules for all odd numbers.
 
I understood ..Thank you all
 
You can totally remove "prime" from that statement. Any odd INTEGER is 1 or 3 mod 4! If it's 0 or 2 mod 4 then it's even. I feel like bouncing my lips with my index finger in a rude gesture now.
 

Similar threads

Undergrad How to Prove a Congruence Relation for Prime Numbers?
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
1K
Undergrad Question on Primitive Roots and GCDs
  • · Replies 2 ·
Replies
2
Views
2K
If ## p ## and ## p^{2}+8 ## are both prime numbers, prove that....
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Localising a non integral domain at a prime
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Fundamental theorem of arithmetic from Jordan-Hölder theorem
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Proof for subgroup -- How prove it is a subgroup of Z^m?
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Frequency of prime number gaps according to (p-1)/(p-2)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
2K
For n>3, show that the integers n, n+2, n+4 cannot all be prime
  • · Replies 27 ·
Replies
27
Views
3K