diff --git a/src/main/java/org/apache/datasketches/theta/JaccardSimilarity.java b/src/main/java/org/apache/datasketches/theta/JaccardSimilarity.java
index 2fc9d4fe5..f35b246fb 100644
--- a/src/main/java/org/apache/datasketches/theta/JaccardSimilarity.java
+++ b/src/main/java/org/apache/datasketches/theta/JaccardSimilarity.java
@@ -61,28 +61,8 @@ public static double[] jaccard(final ThetaSketch sketchA, final ThetaSketch sket
     if (sketchA.isEmpty() && sketchB.isEmpty()) { return ONES.clone(); }
     if (sketchA.isEmpty() || sketchB.isEmpty()) { return ZEROS.clone(); }
 
-    final int countA = sketchA.getRetainedEntries(true);
-    final int countB = sketchB.getRetainedEntries(true);
-
-    //Create the ThetaUnion
-    final int minK = 1 << ThetaUtil.MIN_LG_NOM_LONGS;
-    final int maxK = 1 << ThetaUtil.MAX_LG_NOM_LONGS;
-    final int newK = max(min(ceilingPowerOf2(countA + countB), maxK), minK);
-    final ThetaUnion union =
-        ThetaSetOperation.builder().setNominalEntries(newK).buildUnion();
-    union.union(sketchA);
-    union.union(sketchB);
-    final ThetaSketch unionAB = union.getResult(false, null);
-    final long thetaLongUAB = unionAB.getThetaLong();
-    final long thetaLongA = sketchA.getThetaLong();
-    final long thetaLongB = sketchB.getThetaLong();
-    final int countUAB = unionAB.getRetainedEntries(true);
-
-    //Check for identical data
-    if (countUAB == countA && countUAB == countB
-        && thetaLongUAB == thetaLongA && thetaLongUAB == thetaLongB) {
-      return ONES.clone();
-    }
+    final ThetaSketch unionAB = computeUnion(sketchA, sketchB);
+    if (identicalSets(sketchA, sketchB, unionAB)) { return ONES.clone(); }
 
     //Create the ThetaIntersection
     final ThetaIntersection inter = ThetaSetOperation.builder().buildIntersection();
@@ -100,6 +80,10 @@ public static double[] jaccard(final ThetaSketch sketchA, final ThetaSketch sket
   /**
    * Returns true if the two given sketches have exactly the same hash values and the same
    * theta values. Thus, they are equivalent.
+   *
+   * <p>Note: For very large pairs of sketches, where the configured nominal entries of the sketches
+   * are 2^26, this method may return false even if the two sketches are equal.
+   *
    * @param sketchA the given sketch A
    * @param sketchB the given sketch B
    * @return true if the two given sketches have exactly the same hash values and the same
@@ -112,26 +96,8 @@ public static boolean exactlyEqual(final ThetaSketch sketchA, final ThetaSketch
     if (sketchA.isEmpty() && sketchB.isEmpty()) { return true; }
     if (sketchA.isEmpty() || sketchB.isEmpty()) { return false; }
 
-    final int countA = sketchA.getRetainedEntries(true);
-    final int countB = sketchB.getRetainedEntries(true);
-
-    //Create the ThetaUnion
-    final ThetaUnion union =
-        ThetaSetOperation.builder().setNominalEntries(ceilingPowerOf2(countA + countB)).buildUnion();
-    union.union(sketchA);
-    union.union(sketchB);
-    final ThetaSketch unionAB = union.getResult();
-    final long thetaLongUAB = unionAB.getThetaLong();
-    final long thetaLongA = sketchA.getThetaLong();
-    final long thetaLongB = sketchB.getThetaLong();
-    final int countUAB = unionAB.getRetainedEntries(true);
-
-    //Check for identical counts and thetas
-    if (countUAB == countA && countUAB == countB
-        && thetaLongUAB == thetaLongA && thetaLongUAB == thetaLongB) {
-      return true;
-    }
-    return false;
+    final ThetaSketch unionAB = computeUnion(sketchA, sketchB);
+    return identicalSets(sketchA, sketchB, unionAB);
   }
 
   /**
@@ -178,4 +144,43 @@ public static boolean dissimilarityTest(final ThetaSketch measured, final ThetaS
     return jRatioUB <= threshold;
   }
 
+  /**
+   * Computes the union of the two given sketches. The union is sized to hold the retained entries
+   * of both sketches, clamped to the legal range of nominal entries. The result is unordered,
+   * because only its retained entries and theta are used.
+   * @param sketchA the given sketch A
+   * @param sketchB the given sketch B
+   * @return the unordered union of sketch A and sketch B
+   */
+  private static ThetaSketch computeUnion(final ThetaSketch sketchA, final ThetaSketch sketchB) {
+    final int countA = sketchA.getRetainedEntries(true);
+    final int countB = sketchB.getRetainedEntries(true);
+    final int minK = 1 << ThetaUtil.MIN_LG_NOM_LONGS;
+    final int maxK = 1 << ThetaUtil.MAX_LG_NOM_LONGS;
+    final int newK = max(min(ceilingPowerOf2(countA + countB), maxK), minK);
+    final ThetaUnion union =
+        ThetaSetOperation.builder().setNominalEntries(newK).buildUnion();
+    union.union(sketchA);
+    union.union(sketchB);
+    return union.getResult(false, null);
+  }
+
+  /**
+   * Returns true if the two given sketches and their union all have the same number of retained
+   * entries and the same theta, which means the two sketches hold identical sets.
+   * @param sketchA the given sketch A
+   * @param sketchB the given sketch B
+   * @param unionAB the union of sketch A and sketch B
+   * @return true if the two given sketches hold identical sets
+   */
+  private static boolean identicalSets(final ThetaSketch sketchA, final ThetaSketch sketchB,
+      final ThetaSketch unionAB) {
+    final int countUAB = unionAB.getRetainedEntries(true);
+    final long thetaLongUAB = unionAB.getThetaLong();
+    return countUAB == sketchA.getRetainedEntries(true)
+        && countUAB == sketchB.getRetainedEntries(true)
+        && thetaLongUAB == sketchA.getThetaLong()
+        && thetaLongUAB == sketchB.getThetaLong();
+  }
+
 }
diff --git a/src/test/java/org/apache/datasketches/theta/JaccardSimilarityTest.java b/src/test/java/org/apache/datasketches/theta/JaccardSimilarityTest.java
index 4f84c5382..17cadcc92 100644
--- a/src/test/java/org/apache/datasketches/theta/JaccardSimilarityTest.java
+++ b/src/test/java/org/apache/datasketches/theta/JaccardSimilarityTest.java
@@ -233,6 +233,17 @@ public void checkMinK() {
     println(result[0] + ", " + result[1] + ", " + result[2]);
   }
 
+  @Test
+  public void checkMinKExactlyEqual() {
+    UpdatableThetaSketch skA = UpdatableThetaSketch.builder().build(); //4096
+    UpdatableThetaSketch skB = UpdatableThetaSketch.builder().build(); //4096
+    skA.update(1);
+    skB.update(1);
+    assertTrue(exactlyEqual(skA, skB));
+    skB.update(2);
+    assertFalse(exactlyEqual(skA, skB));
+  }
+
   @Test
   public void printlnTest() {
     println("PRINTING: "+this.getClass().getName());