Technical Task
Fast Positional Decomposition of Large Numbers
1. PROBLEM STATEMENT
Goal: Develop a system for fast positional decomposition of numbers ranging from 6 billion to 30 billion digits with subsequent precise mathematical reconstruction.
2. INPUT DATA CHARACTERISTICS
2.1 Number Structure
- Size: From 6 billion to 30 billion digits
- Digit Base: The number consists exclusively of digits
1, 2, 3, 4 - Format: Continuous sequence of digits without separators
- Example:
12341234123412341234... (6-30 billion digits)
2.2 Constraints and Requirements
- Accuracy: 100% mathematical accuracy of reconstruction without loss
- Speed: Decomposition must be performed in milliseconds (seconds maximum)
- Result: 4-8 numerical coefficients
3. POSITIONAL DECOMPOSITION ALGORITHM
3.1 Mathematical Model
Principle of Sequential Division:
The input number N is decomposed according to the formula:
N = K₁×B₁ + K₂×B₂ + K₃×B₃ + K₄×B₄ + K₅×B₅
Where:
N - initial number (6-30 billion digits)K₁, K₂, K₃, K₄, K₅ - decomposition coefficientsB₁, B₂, B₃, B₄, B₅ - positional bases (divisors)
3.2 Step-by-Step Decomposition Algorithm
Step 1: Division by the largest base
N ÷ B₁ = K₁ (quotient) + R₁ (remainder)
B₁ = 100,000,000 (100 million)
Step 2: Division of the remainder by the second base
R₁ ÷ B₂ = K₂ (quotient) + R₂ (remainder)
B₂ = 100,000 (100 thousand)
Step 3: Continuation of decomposition
R₂ ÷ B₃ = K₃ + R₃, where B₃ = 1,000
R₃ ÷ B₄ = K₄ + R₄, where B₄ = 10
R₄ ÷ B₅ = K₅ + 0, where B₅ = 1
Result: Set of coefficients [K₁, K₂, K₃, K₄, K₅]
3.3 Reconstruction Formula
Exact reconstruction of the original number:
N = K₁×100,000,000 + K₂×100,000 + K₃×1,000 + K₄×10 + K₅×1
4. TECHNICAL REQUIREMENTS FOR SPEED
4.1 Target Performance Indicators
For a number of 6 billion digits:
- Decomposition: ≤ 60 sec max
- Reconstruction: ≤ 30-60 max
- Full cycle: 2-3 min max
Main Problem: Standard algorithms for dividing extremely large numbers are too slow
Necessary Optimizations
8. TESTING AND VERIFICATION
8.1 Test Data Sets
Small Tests (up to 1 million digits):
- Full verification of all methods
- Comparison with reference results
- Testing edge cases
Large Tests (6-30 billion digits):
- Performance testing
- Resource usage verification
- Stability stress testing
8.2 Quality Criteria
Functionality:
- ✅ 100% mathematical accuracy of reconstruction
- ✅ Support for numbers up to 30 billion digits
- ✅ Stable operation under repeated use
10. EXPECTED RESULTS
10.1 Technical Achievements
- Speed: Decomposition of numbers in seconds
- Mathematical Accuracy: 100% lossless reconstruction
Essentially, a fast division needs to be developed
Tests will be conducted on standard laptops like Dell Latitude 5400 (32GB RAM)
