The thought struck me as to what primes in base 10 remain prime when the digits stay the same but the number base increases to 11, 12, 13 etc. I got Gemini to investigate and it came up with useful summary and an impressive interactive tool for investigation:
To analyze this integer sequence mathematically, we can define the transformation as evaluating a polynomial.
Let a base-10 prime \( p \) have the decimal representation \( d_k d_{k-1} \dots d_1 d_0 \). The value of \( p \) is given by:
\[ p = \sum_{i=0}^{k} d_i 10^i \]When we take these same digits and interpret them in a higher base \( b \) (where \( 11 \le b \le 36 \)), we are evaluating the new integer \( P_b \):
\[ P_b = \sum_{i=0}^{k} d_i b^i \]The task is to find all instances where both \( p \) and \( P_b \) are prime. When we run this analysis across different bases, distinct number-theoretic patterns emerge, particularly regarding parity and divisibility.
The Parity Sieve for Odd Bases
The parity of the new number \( P_b \) depends entirely on whether the target base \( b \) is odd or even.
If the target base \( b \) is odd (e.g., base 11, 13, 15), then \( b \equiv 1 \pmod 2 \). Consequently, the modulo-2 value of the new number collapses to the sum of its digits:
\[ P_b \equiv \sum_{i=0}^{k} d_i \pmod 2 \]This creates a massive sieve: if the sum of the digits of the original prime \( p \) is even, \( P_b \) will inevitably be even. Since \( P_b > 2 \) for all target primes in these higher bases, an even \( P_b \) is strictly composite. Therefore, odd bases yield a significantly sparser sequence of primes.
The Density in Even Bases
If the target base \( b \) is even (like the default \( b=16 \)), then \( b \equiv 0 \pmod 2 \). In this scenario, all terms multiplied by \( b \) are even, meaning the parity of \( P_b \) relies solely on the final digit \( d_0 \):
\[ P_b \equiv d_0 \pmod 2 \]Because \( p \) is prime in base-10, its final digit \( d_0 \) must be odd (1, 3, 7, or 9) with the sole exception of \( p=2 \). Therefore, \( P_b \) will always be odd in an even base. Because they bypass the digit-sum parity trap, even bases like 16 retain a much higher density of prime numbers.
Small Examples in Base 16
For \( b=16 \), examining the first few base-10 primes reveals the retention rate:
- 11: \( 1(16) + 1 = 17 \) (Prime)
- 13: \( 1(16) + 3 = 19 \) (Prime)
- 17: \( 1(16) + 7 = 23 \) (Prime)
- 19: \( 1(16) + 9 = 25 \) (Composite, \( 5 \times 5 \))
- 23: \( 2(16) + 3 = 35 \) (Composite, \( 5 \times 7 \))
- 29: \( 2(16) + 9 = 41 \) (Prime)
To thoroughly investigate the sequence up to 40,000 across any base from 11 to 36 without relying on a static, truncated list, the interactive tool below is designed to compute the results dynamically. It handles the base conversions and primality testing locally within your browser.
