Number Theory: An Introduction to Mathematics

3 Multiplicative Functions 159

Suppose first thatMpdividesSp− 1 and assume thatMpis composite. Ifqis the
least prime divisor ofMp,thenq^2 ≤Mpandq=2. By hypothesis,

ω^2

p− 2

+ω′^2

p− 2

≡0modq.

Now considerωandω′as elements ofJq. By multiplying byω^2
p− 2
, we obtain

ω^2
p− 1
=−1 and henceω^2
p
=1. Thusω∈Jq×and the order ofωinJq×is exactly 2p.
Hence

2 p≤q^2 − 1 ≤Mp− 1 = 2 p− 2 ,

which is a contradiction.
Suppose next thatMp=qis prime. Thenq≡−1 mod 8, sincep≥3. Since

( 2 /q)=(− 1 )(q

(^2) − 1 )/ 8
, it follows that 2 is a quadratic residue ofq. Thus there exists an
integerasuch that
a^2 ≡2modq.
Furthermoreq ≡1 mod 3, since 2^2 ≡1 and hence 2p−^1 ≡1 mod 3. Thusqis a
quadratic residue of 3. Sinceq ≡−1 mod 4, it follows from the law of quadratic
reciprocity that 3 is a quadratic nonresidue ofq. Hence, by Euler’s criterion (Proposi-
tion II.28),
3 (q−^1 )/^2 ≡−1modq.
Consider the elementτ=aq−^2 ( 1 +

√

3 )ofJq.Wehave

τ^2 = 2 q−^2 · 2 ω=ω,

since 2q−^1 ≡1modq. On the other hand,

( 1 +

√

3 )q= 1 + 3 (q−^1 )/^2

√

3 = 1 −

√

3

and hence

τq=aq−^2 ( 1 −

√

3 ).

Consequently,

ω(q+^1 )/^2 =τq+^1 =aq−^2 ( 1 −

√

3 )·aq−^2 ( 1 +

√

3 )= 2 q−^2 (− 2 )=− 1.

Multiplying byω′(q+^1 )/^4 , we obtainω(q+^1 )/^4 =−ω′(q+^1 )/^4. In other words, since
(q+ 1 )/ 4 = 2 p−^2 ,

Sp− 1 =ω^2

p− 2

+ω′^2

p− 2

≡0modq. 

It is conjectured that there are infinitely many Mersenne primes, and hence infi-
nitely many even perfect numbers. A heuristic argument of Gillies (1964), as modified
by Wagstaff (1983), suggests that the number of primesp≤xfor whichMpis prime
is asymptotic to(eγ/log 2)logx,whereγis Euler’s constant (Chapter IX,§4) and
thuseγ/log 2= 2. 570 ....