SageMath makes working with integers super intuitive — from
checking primality to exploring number theory!
Today, we dive into SageMath tools that make integer exploration easy, visual,
and real-world-ready!
1. Basic Integer Operations
- Checks
if n is an integer.
Is n prime?
Factorize n:
Find next and previous primes:
Find divisors of n:
2. Digging into Digits
Bigger Number Example:
3. Division and Remainders ➗
Example:
๐ Visual Reinforcement:
Division Formula
|
Variable
|
Value
|
|
a
|
7642
|
|
b
|
63
|
|
q
|
121
|
|
r
|
19
|
Check in SageMath:
4. GCD and Extended GCD ๐
Finding GCD:
Extended GCD (Bezout's Identity):
๐ GCD Process Flowchart:
5. Real-World Cryptography with SageMath ๐
Encrypt a Simple Message:
Decrypt the Message:
๐ How Key and Modulus are
Derived:
- Choose
two primes
- Calculate
modulus:
- Calculate
- Choose
encryption key
such that 
- Compute
decryption key
6. Prime Numbers and Euler’s Phi Function ๐ข
Counting Primes:
Euler’s Phi Function:
7. Huge Numbers: Factorials ๐
Calculating and Exploring Factorials:
Analyzing Digits:
8. Real Numbers and Approximations
9. Working with Arrays (NumPy + SageMath)
✍️ Practice Exercises
|
Exercise
|
Hint
|
|
Find LCM and verify
relation with GCD
|
Use
|
|
Extended GCD for three integers
|
Apply 
|
|
Factors of sum of
digits of 
|
Use 
|
|
Count integers coprime to 12 between 62 and 672
|
Use a simple for
loop with 
|
|
Find n such that 
|
Try 
|
|
Compute Euler’s phi function manually
|
Loop and
count with 
|
⚡ Efficiency Highlights
- SageMath
handles huge numbers (factorials, big primes) effortlessly.
- Backed
by GMP, MPFR, FLINT libraries for high-speed computation.
- Practical
for cryptography, combinatorics, and real-world number theory.
๐ Join the Learning
Community
๐ฏ Your Challenge:
Solve exercises, create your own SageMath projects, or explore a new number
theory concept!
๐ฌ Share your work:
Post your GitHub repository links or paste SageMath code snippets in the
comments!
๐ฎ What’s Next?
Get ready for upcoming posts:
- Symbolic
Computation: Solving algebraic equations and calculus in SageMath
- Graph
Theory: Building graphs and analyzing networks
- Cryptography
Toolkit: Generate your own RSA keys
- Advanced
Number Theory: Dive into Pell’s Equation and Elliptic Curves
๐ Conclusion
SageMath transforms working with integers into an exciting
adventure — from basic division to powerful cryptographic applications!
Through visualization, exploration, and real-world connections, you can master
computational mathematics one step at a time.
Keep Exploring, Keep Creating! ๐
๐ Next Challenge Topic
Preview
"Solving Equations in SageMath"
SageMath isn't just about numbers—it's a powerhouse for
solving equations! In this challenge, you'll explore everything from basic
algebraic equations to advanced systems of equations. Highlights include:
- Single-Variable
Equations
Like: Solve (x^2 - 5x + 6 = 0).
- Systems
of Linear Equations
Solve real-world problems with equations like (2x + 3y = 5) and (x - y =
1).
- Non-linear
Equations and Applications
Dive into quadratic, exponential, and trigonometric equations.
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