Solving a Number Theory Problem Using Fermat's Little Theorem

In summary, Number theory problem is a branch of mathematics that deals with the properties and relationships of numbers. Some common topics include prime numbers, divisibility, modular arithmetic, and Diophantine equations. It can be challenging due to its abstract concepts and open-ended problems. Number theory is used in various practical applications such as computer science and cryptography. Some famous unsolved problems in number theory include the Goldbach conjecture, the Riemann hypothesis, and the Twin prime conjecture.
  • #1
ehrenfest
2,020
1

Homework Statement


http://math.stanford.edu/~vakil/putnam07/07putnam2.pdf

I am working on number 2.
So I want to find 2^70 + 3^70 mod 13.
I can use Fermat's Little Theorem to reduce the exponent to 10, but I do not know what to do next...


Homework Equations





The Attempt at a Solution

 
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  • #2
2^2 = 4 and 3^2 = 9

and 4^5 + 9^5 = (4+9)*something.
 
  • #3
morphism said:
4^5 + 9^5 = (4+9)*something.
Is that true? Where does that come from?
 
  • #4
You know how there's a formula for a^n - b^n? Well, there's also one for a^n + b^n when n is odd. (a^n + b^n = a^n - (-b)^n.)
 
  • #5
I see. Thanks.
 

What is number theory problem?

Number theory problem is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves studying patterns and structures within numbers, as well as solving problems and proofs related to number theory.

What are some common topics in number theory problem?

Some common topics in number theory problem include prime numbers, divisibility, modular arithmetic, and Diophantine equations. Other topics include number sequences, perfect numbers, and the distribution of primes.

What makes number theory problem challenging?

Number theory problem can be challenging because it often involves abstract concepts and complex patterns. Additionally, many number theory problems are open-ended and require creative thinking and problem-solving skills to solve.

How is number theory problem used in real life?

Number theory problem has many practical applications in fields such as computer science, cryptography, and physics. For example, number theory is used in developing encryption algorithms for secure communication and in designing efficient computer algorithms.

What are some famous unsolved problems in number theory?

Some famous unsolved problems in number theory include the Goldbach conjecture, the Riemann hypothesis, and the Twin prime conjecture. These problems have been open for many years and have sparked much interest and research in the field.

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