diff --git a/junk/faulhaber/faulhaber.html b/junk/faulhaber/faulhaber.html
index fc5ab78..25d1baf 100644
--- a/junk/faulhaber/faulhaber.html
+++ b/junk/faulhaber/faulhaber.html
@@ -20,6 +20,7 @@
 				<hgroup>
 					<h1>Sums of Powers</h1>
 					<p>Posted <time datetime="2026-04-15">April 15, 2026</time></p>
+					<p>Updated <time datetime="2026-06-10">June 10, 2026</time></p>
 				</hgroup>
 
 				<p>This page is all about how we can efficiently compute large sums of
@@ -402,7 +403,7 @@
 
 						<p>This algorithm is quadratic in the value of
 						<math><mi>p</mi></math>, which isn't great, but it's simple to
-						implement (and I don't know of any faster way).</p>
+						implement.<sup>1</sup></p>
 
 						<section>
 							<h4>Stirling numbers calculator</h4>
@@ -460,7 +461,7 @@
 
 						<p>It turns out there is a very elegant identity relating
 						exponents, Stirling numbers, and falling
-						factorials:<sup>1</sup></p>
+						factorials:<sup>2</sup></p>
 						<div class="math-block">
 						<math display="block">
 							<mrow>
@@ -836,8 +837,8 @@
 						</math>
 						</div>
 						<p>(For <math><mi>p</mi><mo>=</mo><mn>0</mn></math>, the sum is
-						just <math><mi>n</mi></math>.)<sup>2</sup> This lets us
-						simplify.<sup>3</sup></p>
+						just <math><mi>n</mi></math>.)<sup>3</sup> This lets us
+						simplify.<sup>4</sup></p>
 
 						<div class="math-block">
 						<math display="block">
@@ -1057,13 +1058,21 @@
 				<section>
 					<h2>Footnotes</h2>
 					<ol id="footnote-list">
-						<li><sup>1</sup> For proofs of this identity, see chapter 1.9 of
+						<li><sup>1</sup> I think we can technically compute all the Stirling
+							numbers we need in linearithmic time by applying the convolution
+							theorem to the numbers' explicit sum formula, but that approach
+							makes it difficult to get exact values. If you just want the
+							Stirling numbers modulo a prime, you could maybe use a
+							<a href="https://en.wikipedia.org/wiki/Discrete_Fourier_transform_over_a_ring">number-theoretic transform</a>,
+							but that's well outside my wheelhouse so I'm not going to touch
+						on that here.</li>
+						<li><sup>2</sup> For proofs of this identity, see chapter 1.9 of
 							<cite>Enumerative Combinatorics</cite> by Stanley and chapter 6.1
 							of <cite>Concrete Mathematics</cite> by Graham, Knuth, and
 						Patashnik.</li>
-						<li><sup>2</sup> See chapter 2.6 of <cite>Concrete
+						<li><sup>3</sup> See chapter 2.6 of <cite>Concrete
 								Mathematics</cite> for an explanation of this identity.</li>
-						<li><sup>3</sup> A version of this formula is also mentioned in
+						<li><sup>4</sup> A version of this formula is also mentioned in
 							chapter 6.5 of <cite>Concrete Mathematics</cite>.</li>
 					</ol>
 				</section>