From 2e8234232da3e6d0ffeccaea081810862b3e6f99 Mon Sep 17 00:00:00 2001
From: Peter Norvig <peter.norvig@gmail.com>
Date: Sun, 29 Mar 2026 19:44:23 -0700
Subject: [PATCH] Update explanation of modular exponentiation

---
 ipynb/TruncatablePrimes.ipynb | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ipynb/TruncatablePrimes.ipynb b/ipynb/TruncatablePrimes.ipynb
index 7ecf305..323df2e 100644
--- a/ipynb/TruncatablePrimes.ipynb
+++ b/ipynb/TruncatablePrimes.ipynb
@@ -1127,7 +1127,7 @@
     "\n",
     "# Note on Modular Exponentiation\n",
     "\n",
-    "Just one more thing: none of this would work unless we can efficiently compute *a*<sup>(*n* - 1)</sup> (mod *n*).  How does the `pow` builtin function do it? When *a* and *n* are 24-digit numbers, if we naively tried to compute `a ** (n - 1)`, we'd have two problems: we'd need nearly a billion petabytes to store the result, and we'd need centuries to compute it. The way around these problems is to use [modular exponentiation](https://en.wikipedia.org/wiki/Modular_exponentiation) where we apply the modulus to each intermediate result, and cut the exponent in half each iteration, so we need only do *O*(log *n*) multiplications, not *O*(*n*). That's a big difference: 10<sup>24</sup> is a trillion trillion, and log<sub>2</sub>(10<sup>24</sup>) is only 80. \n",
+    "Just one more thing: none of this would work unless we can efficiently compute *a*<sup>(*n* - 1)</sup> (mod *n*).  How does the `pow` builtin function do it? When *a* and *n* are 24-digit numbers, if we naively tried to compute `a ** (n - 1)` and then apply (mod *n*), we'd have two problems: we'd need nearly a billion petabytes to store the result, and we'd need centuries to compute it. The way around these problems is to use [modular exponentiation](https://en.wikipedia.org/wiki/Modular_exponentiation) where we apply the modulus to each intermediate result, and cut the exponent in half each iteration, so we need only do *O*(log *n*) multiplications, not *O*(*n*). That's a big difference: 10<sup>24</sup> is a trillion trillion, and log<sub>2</sub>(10<sup>24</sup>) is only 80. \n",
     "\n",
     "There are two key ides that make this work:\n",
     "1) *b*<sup>2*e*</sup> = (*b* × *b*)<sup>*e*</sup>\n",