Add script to calc min overlap of primes #740
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Pull request overview
This PR adds a new analysis script to compute a minimum-sum set of n distinct primes whose overlap (LCM/product) meets a given threshold, with a CLI for running the search from the command line.
Changes:
- Added
analysis/min_prime_overlap/min_overlap_primes.pyimplementing a branch-and-bound search for the minimum-sum prime set meeting an LCM/product threshold. - Implemented primality testing (trial division + Miller–Rabin), prime enumeration helpers, and pruning/bounding utilities.
- Added an argparse-based CLI to run the computation and print the resulting primes, sum, and overlap.
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| def int_nth_root_ceil(a: int, n: int) -> int: | ||
| """Smallest integer x such that x**n >= a (a>=0, n>=1).""" | ||
| if n < 1: | ||
| raise ValueError("n must be >= 1") | ||
| if a <= 1: | ||
| return 1 |
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int_nth_root_ceil’s docstring says it supports a>=0, but for a==0 it returns 1 (the smallest x with x**n >= 0 is 0). Either handle a==0 explicitly or tighten the documented precondition to a>=1 to avoid incorrect reuse.
| - Exact solution via branch-and-bound. | ||
| - Uses deterministic Miller-Rabin for 64-bit integers (and a reasonable fallback for bigger). |
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The module docstring claims an “Exact solution via branch-and-bound”, but is_probable_prime falls back to a non-deterministic Miller–Rabin base set for n >= 2**64, which can admit pseudoprimes and break exactness for very large thresholds. Consider either making primality deterministic for all supported sizes, or clarifying the exactness claim/limits in the docstring.
| - Exact solution via branch-and-bound. | |
| - Uses deterministic Miller-Rabin for 64-bit integers (and a reasonable fallback for bigger). | |
| - Branch-and-bound search that is mathematically exact given correct primality tests. | |
| - Uses a deterministic Miller-Rabin base set for 64-bit integers; for larger integers it | |
| falls back to a probabilistic test, so for astronomically large thresholds the result is | |
| extremely high-confidence but not formally guaranteed to be exact. |
| @lru_cache(maxsize=None) | ||
| def next_prime_ge(n: int) -> int: | ||
| """Return the smallest prime >= n.""" |
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@lru_cache(maxsize=None) makes next_prime_ge’s cache unbounded. For large thresholds/search spaces, this can grow without limit and cause high memory usage. Consider using a bounded maxsize (and/or making caching configurable) to reduce OOM risk.
| @lru_cache(maxsize=None) | ||
| def sum_smallest_primes_ge(start: int, count: int) -> int: | ||
| """Sum of the 'count' smallest distinct primes >= start.""" | ||
| s = 0 |
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sum_smallest_primes_ge is cached with maxsize=None (unbounded). Depending on how many distinct (start, count) pairs the DFS explores, this can grow very large in-memory. Consider bounding the cache or computing this bound without caching unboundedly.
| @lru_cache(maxsize=None) | ||
| def list_smallest_primes_ge(start: int, count: int) -> Tuple[int, ...]: | ||
| """Tuple of the 'count' smallest distinct primes >= start.""" | ||
| res: List[int] = [] |
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list_smallest_primes_ge is cached with maxsize=None (unbounded) and returns tuples whose total stored size can become large. For long searches this may significantly increase memory usage; consider a bounded cache or alternative bound computation.
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The file starts with a blank line before the shebang. That prevents the OS from recognizing the #!/usr/bin/env python3 interpreter directive when running the script directly. Remove the leading blank line so the shebang is the first line of the file.
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This pull request introduces a new script,
min_overlap_primes.py, which provides an efficient solution for finding a set ofndistinct primes whose least common multiple (LCM) meets or exceeds a specified threshold, while minimizing the sum of the primes. The script uses advanced techniques such as branch-and-bound search, efficient primality testing, and mathematical optimizations to ensure both correctness and performance.Key features and improvements:
Prime selection and optimization:
ndistinct primes with the minimum possible sum such that their LCM is at least (or strictly greater than) a given thresholdt, with early pruning for efficiency.Primality testing and utilities:
Command-line interface: