From 5a9e103bd50908bbbd4f5b69235b766d32ad0c77 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Lema=C5=84czyk?= <marek@lemanczyk.eu>
Date: Wed, 29 Oct 2025 01:29:12 +0100
Subject: [PATCH] Replace heuristic sqrt with log2-based estimate

---
 .../HeuristicCombinedPrimeTester.cs           | 29 +++++++++----------
 PerfectNumbers.Core/HeuristicPrimeTester.cs   | 29 +++++++++----------
 2 files changed, 26 insertions(+), 32 deletions(-)

diff --git a/PerfectNumbers.Core/HeuristicCombinedPrimeTester.cs b/PerfectNumbers.Core/HeuristicCombinedPrimeTester.cs
index 8d29efc..05c22a8 100644
--- a/PerfectNumbers.Core/HeuristicCombinedPrimeTester.cs
+++ b/PerfectNumbers.Core/HeuristicCombinedPrimeTester.cs
@@ -311,31 +311,28 @@ Cleanup:
             return n;
         }
 
-        ulong bit = 1UL << 62;
-        while (bit > n)
+        int log2 = BitOperations.Log2(n);
+        ulong estimate = 1UL << ((log2 + 1) >> 1);
+        ulong quotient = n / estimate;
+
+        while (estimate > quotient)
         {
-            bit >>= 2;
+            estimate = (estimate + quotient) >> 1;
+            quotient = n / estimate;
         }
 
-        ulong result = 0UL;
-        ulong remainder = n;
-        while (bit != 0)
+        while (true)
         {
-            ulong candidate = result + bit;
-            if (remainder >= candidate)
+            ulong nextEstimate = estimate + 1UL;
+            if (nextEstimate > n / nextEstimate)
             {
-                remainder -= candidate;
-                result = (result >> 1) + bit;
-            }
-            else
-            {
-                result >>= 1;
+                break;
             }
 
-            bit >>= 2;
+            estimate = nextEstimate;
         }
 
-        return result;
+        return estimate;
     }
 
     private static ReadOnlySpan<uint> GetCombinedDivisors(byte nMod10)
diff --git a/PerfectNumbers.Core/HeuristicPrimeTester.cs b/PerfectNumbers.Core/HeuristicPrimeTester.cs
index d8fb2cd..d752147 100644
--- a/PerfectNumbers.Core/HeuristicPrimeTester.cs
+++ b/PerfectNumbers.Core/HeuristicPrimeTester.cs
@@ -436,31 +436,28 @@ Cleanup:
             return n;
         }
 
-        ulong bit = 1UL << 62;
-        while (bit > n)
+        int log2 = BitOperations.Log2(n);
+        ulong estimate = 1UL << ((log2 + 1) >> 1);
+        ulong quotient = n / estimate;
+
+        while (estimate > quotient)
         {
-            bit >>= 2;
+            estimate = (estimate + quotient) >> 1;
+            quotient = n / estimate;
         }
 
-        ulong result = 0UL;
-        ulong remainder = n;
-        while (bit != 0)
+        while (true)
         {
-            ulong candidate = result + bit;
-            if (remainder >= candidate)
+            ulong nextEstimate = estimate + 1UL;
+            if (nextEstimate > n / nextEstimate)
             {
-                remainder -= candidate;
-                result = (result >> 1) + bit;
-            }
-            else
-            {
-                result >>= 1;
+                break;
             }
 
-            bit >>= 2;
+            estimate = nextEstimate;
         }
 
-        return result;
+        return estimate;
     }
 
     private static ReadOnlySpan<byte> GetGroupBEndingOrder(byte nMod10) => nMod10 switch