10. Primes and composites in other sequences 31
The reader should note that the Shapiro–Sparer paper contains several
other attractive results on composite numbers in various sequences.
We close this section by considering the sequence of shifted factorials
n! + 1. Here we can easily obtain infinitely many composite terms, since
Wilson’s theorem implies that (p 1)! + 1 is composite for each p 3. The
following pretty theorem of Schinzel [Sch62b] generalizes this result:
Theorem 1.31. Let α be a positive rational number. Then there are infin-
itely many n for which α · n! + 1 is composite.
Lemma 1.32. Let p be a prime and let r and s be positive integers. Then
for 0 i p 1, we have
p | si! +
(−1)i+1r
⇐⇒ p | r(p 1 i)! + s.
Proof. By Wilson’s theorem,
−1 (p 1)! = (p 1)(p 2) · · · (p i)(p i 1)!

(−1)ii!(p
1 i)! (mod p),
so that (p−1−i)!i!
(−1)i+1
(mod p). Since p and (p−1−i)! are coprime,
p | si! +
(−1)i+1r
⇐⇒ p | s(p 1 i)!i! +
(−1)i+1r(p
1 i)!
⇐⇒ p |
(−1)i+1s
+
(−1)i+1r(p
1 i)!
⇐⇒ p | s + r(p 1 i)!.
Proof of Theorem 1.31. Write α = r/s, where r and s are relatively
prime positive integers. Assume l N and l r/2. Then
(4l)!α−1
is an
integer divisible by both 4 and r. Since 4 |
(4l)!α−1,
we can choose a prime
pl −1 (mod 4) with
pl |
(4l)!α−1
1.
Because r |
(4l)!α−1,
necessarily pl r. Since
(1.13) pl | r
(
(4l)!α−1
1
)
= s(4l)! r,
we must have pl 4l. From Lemma 1.32 (with i = 4l) and (1.13), we find
that
(1.14) pl | r(pl 4l 1)! + s.
Since pl r, (1.14) implies that pl s, and so
pl | Nl := α(pl 4l 1)! + 1
whenever Nl is an integer. This happens for all large l: Indeed, from (1.14)
we have Nl pl/s 4l/s, so that Nl with l, which is only possible if
pl 4l 1 with l. But Nl is an integer whenever pl 4l 1 s.
Finally, notice that for large l, we cannot have pl = Nl, since pl
−1 (mod 4) while Nl 1 (mod 4). Thus Nl is a composite integer of the
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