diff --git a/Tests/test_file_jpeg.py b/Tests/test_file_jpeg.py
index be8e21d5a..1801cd8ff 100644
--- a/Tests/test_file_jpeg.py
+++ b/Tests/test_file_jpeg.py
@@ -3,7 +3,14 @@ import re
 from io import BytesIO
 
 import pytest
-from PIL import ExifTags, Image, ImageFile, JpegImagePlugin, features
+from PIL import (
+    ExifTags,
+    Image,
+    ImageFile,
+    JpegImagePlugin,
+    UnidentifiedImageError,
+    features,
+)
 
 from .helper import (
     assert_image,
@@ -709,6 +716,24 @@ class TestFileJpeg:
         with Image.open("Tests/images/icc-after-SOF.jpg") as im:
             assert im.info["icc_profile"] == b"profile"
 
+    def test_jpeg_magic_number(self):
+        size = 4097
+        buffer = BytesIO(b"\xFF" * size)  # Many xFF bytes
+        buffer.max_pos = 0
+        orig_read = buffer.read
+
+        def read(n=-1):
+            res = orig_read(n)
+            buffer.max_pos = max(buffer.max_pos, buffer.tell())
+            return res
+
+        buffer.read = read
+        with pytest.raises(UnidentifiedImageError):
+            Image.open(buffer)
+
+        # Assert the entire file has not been read
+        assert 0 < buffer.max_pos < size
+
 
 @pytest.mark.skipif(not is_win32(), reason="Windows only")
 @skip_unless_feature("jpg")
diff --git a/src/PIL/JpegImagePlugin.py b/src/PIL/JpegImagePlugin.py
index 89e70f0e9..6edaee795 100644
--- a/src/PIL/JpegImagePlugin.py
+++ b/src/PIL/JpegImagePlugin.py
@@ -323,7 +323,8 @@ MARKER = {
 
 
 def _accept(prefix):
-    return prefix[0:1] == b"\377"
+    # Magic number was taken from https://en.wikipedia.org/wiki/JPEG
+    return prefix[0:3] == b"\xFF\xD8\xFF"
 
 
 ##
@@ -337,10 +338,11 @@ class JpegImageFile(ImageFile.ImageFile):
 
     def _open(self):
 
-        s = self.fp.read(1)
+        s = self.fp.read(3)
 
-        if i8(s) != 255:
+        if s != b"\xFF\xD8\xFF":
             raise SyntaxError("not a JPEG file")
+        s = b"\xFF"
 
         # Create attributes
         self.bits = self.layers = 0