Prime Digit Replacements

leetcode
programming
An interesting Prime problem
Author
Affiliation

Kim Hung Bui

Project Euler

Published

July 16, 2025

1 Problem

By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.

2 Analysis

The problem asks for the smallest prime that belongs to an 8-prime value family.

This means: - Start with a prime number. - Replace one or more digits (not necessarily adjacent) that are the same, with a wildcard symbol like *. - Then, generate all possible numbers by replacing * with digits 0 through 9. - Count how many of those are prime. - We want the first (smallest) such prime that produces at least 8 primes in this replacement process.

2.0.1 Why the Solution Space Is Finite

Although large, the space of numbers we need to search is finite because: - We’re looking for the smallest such prime. - The number of primes in each digit-length range is countable.

2.0.3 Strategy Summary

This is essentially a targeted brute-force solution: - Try each prime one by one. - For each repeated digit, try all subsets of its positions. - Replace those positions with digits 0 to 9, checking for primality. - Keep track of how many primes are generated in that “family”. - As soon as we find a family with 8 or more primes, we return the smallest member.

This systematic pruning makes the brute-force approach feasible.

3 Naive Approach - Prime Digit Replacement Problem

3.1 Problem Recap

We want to find the smallest prime number that, by replacing part of its digits (not necessarily adjacent) with the same digit, produces at least eight primes among the 10 generated numbers.

3.2 Naive Strategy

The problems explicitly mentions:

“the 5-digits number 56003 is the first example having seven primes…”

This tell us:

  • Smaller numbers (1- to 4-digit) do not produce a 7-prime family
  • So the search for 8-prime families must begin at least from 5-digit numbers

  1. Loop through numbers starting from 10,000 (since 5-digit numbers are known to be involved).

  2. For each number:

  • Generate all combinations of digit positions to replace (1 or more digits).
  • For each combination:
    • Replace those positions with digits 0 → 9, one at a time.
    • Count how many of the resulting 10 numbers are prime.
  • If ≥ 8 are prime → we’ve found our answer.

3.3 Pseudo code

for number in range(10000, 1000000):
    if not is_prime(number):
        continue

    digits = str(number)
    for mask in all possible combinations of positions:
        family = []
        for replacement_digit in '0123456789':
            candidate_digits = digits with mask replaced by replacement_digit
            if candidate starts with '0':
                continue
            if is_prime(candidate):
                family.append(candidate)
        if len(family) >= 8:
            return number

3.4 Characteristics of Naive Approach

Property Comment
Brute-force Tries every combination of positions and digits
Inefficient Loops over all numbers (not just primes)
No digit filtering May replace digits that occur only once
Replaces non-equal digits Ignores structure of the problem
Slow Explores huge space with many invalid candidates

3.5 Complexity (Rough Estimate)

Let n be the number of digits (5-6 digits):

  • Combinations of digit positions: \(2^n-1\)
  • Replacements: 10 per pattern
  • Prime checks: up to \(10 \times (2^n - 1)\) per number

Overall complexity

\[ O(N\cdot2^n\cdot10\cdot\text{is\_prime}) \]

where N is the range of numbers tested (\(10^6\))

3.6 Problem with Naive

  • Wastes time replacing digits that occur only once
  • Replaces different digits at once (which isn’t allowed per the problem)
  • Replaces digits even when they don’t repeat
  • Explores non-primes too

4 Structured Brute Force Approach

4.1 Motivation

The naive approach checks all numbers and all possible digit replacements. But this is very slow.

A more efficient method is to:

  • Check only primes
  • Replace only repeated digits
  • Ensure no replacements create invalid numbers (like leading zero)
  • Stop early when we find the first family with 8 primes

This gives us a structured brute-force solution.

4.2 Key Insights Over Naive Approach

Improvement Why It Helps
Only check primes Non-primes can’t be part of the answer
Only replace repeated digits You must replace the same digit (e.g. all 1s or all 3s)
Replace some (not all) of the repeated digits Full replacement may cause invalid numbers (e.g. leading zero)
Avoid replacement that starts with 0 Cannot have leading zero in base-10 integers

4.3 Algorithm

4.3.1 Step 1: Generate All Primes Up to a Limit

Use a sieve to get all primes up to a reasonable upper bound (e.g., 1 million).

4.3.2 Step 2: For each Prime

  1. Convert to String
  2. For each digit d (from '0' to '9'):
  • Get positions where d appears more than once
  • Generate all non-empty subsets of those positions
  • Replace those positions with every digit 0–9
  • For each subset:
    • Replace those digits with value 0-9
    • Skip if the number starts with '0'
    • Count how many of the resulting numbers are prime
    • If 8 or more primes found, return the original prime

4.4 Complexity

Let:

  • \(P\): number of primes (~ \(10^5\))
  • \(D\): number of distinct digits (10)
  • \(S\): number of non-empty subsets per digit (usually \(\leq 16\))
  • \(R\): replacements per subset (10 digits)

Then:

\[ O(P\cdot D \cdot S \cdot R) \]

4.5 Summary

Feature Value
Improves over naive Yes (skips non-primes, invalid cases)
Fast enough in practice Yes
Understandable Yes
Prepares for deeper ideas Yes

5 Formal Modeling and Optimization

5.1 Goal

Find the smallest prime p such that replacing a subset of its digits (not necessarily adjacent) with the same digit d, yields a family of 8+ prime numbers out of the 10 possible values.

We aim to go beyond brute-force by modeling the structure of this problem as a constrained optimization over symbolic patterns.

5.2 Step 1: Abstract the Problem as a Search over Equivalence Classes

We define:

  • A prime template: a pattern like '56**3' where * is a wildcard
  • A family function: \[ F(t) = \{ \text{fill in } * \text{ in } t \text{ with digits } 0..9 \} \]

We want:

  • \(|F(t) \cap \text{PRIMES}| \geq 8\)
  • and to find the template t with smallest base value min(F(t)) meeting the above

5.3 Step 2: Symbolic Representation of Digit Replacements

Let each prime p be treated as a string s, and let S ⊂ \text{positions}(s).

We define a mask vector m:

  • m[i] = 1 if index i is to be replaced
  • m[i] = 0 otherwise

Now define a pattern class: \[ T_{p,m} = \{ \text{replace } s[i] \text{ where } m[i]=1 \text{ with } d \in 0..9 \} \]

So the family is generated by iterating over all d ∈ [0,9].

5.4 Step 3: Model the Problem as an Optimization

Variables:

  • Prime p
  • Digit mask m

Constraints:

  • Positions masked must contain the same digit
  • Leading digit after replacement ≠ 0
  • At least 8 of the 10 generated numbers are prime

Objective:

  • Minimize p such that: \[ \left| F(p, m) \cap \text{PRIMES} \right| \geq 8 \]

5.5 Step 4: Optimizations Based on Number Theory

1. Digit Frequency Filtering

  • Only consider digits that repeat at least twice in the prime
  • Avoid replacing rare digits

2. Modulo Pruning

  • Skip candidates that will generate multiples of 2 or 5 in replacement

3. Avoid Unnecessary Masks

  • Skip full replacements
  • Skip replacements that break positional value (e.g., fixed ending 1, 3, 7, 9 often more promising)

5.6 Step 5: Search as Constrained Enumeration

We now enumerate over primes:

  • For each prime, extract repeated digit positions
  • For each digit d, generate all valid mask subsets where digit d occurs
  • Build replacement families
  • Use fast primality tests, and stop early if a mask yields ≥ 8 primes

This avoids full combinatorial enumeration, and uses structure + constraint filtering to reduce search space.

5.7 Theoretical Complexity

Compared to brute force:

Step Complexity
Prime generation \(O(n \log \log n)\)
Mask enumeration Reduced from \(2^k\) to only valid repeated-digit subsets
Replacement testing \(10 \times k\) per valid mask
Primality test \(O(\sqrt{n})\) or Miller–Rabin in \(O(\log^3 n)\)

→ Net result: Much faster and scalable, with full mathematical transparency.

5.8 PhD-Level Interpretation

This transforms the problem into:

A symbolic family detection problem over the space of integers, using digit-based constraints and algebraic structure.

Possible future work includes: - Pattern mining via SAT/SMT solvers - Probabilistic modeling of family distributions - Generalizing to k-prime families

5.9 Summary

Feature PhD Model
Abstraction Symbolic patterns over digits
Constraints Digit repetition, prime count, no leading 0
Objective Minimize base prime p
Tools Digit masks, primality testing, search space pruning
Outcome Structured, fast, extensible method

This approach not only solves the problem but provides a reusable framework for other “digit-replacement” or “prime-pattern” problems in computational number theory.