C O L L O Q U I U M M A T H E M A T I C U M
VOL. LXII 1991 FASC. 2
A NOTE ON PRIMES p WITH σ(pm) = zn
BY
MAOHUA L E (CHANGSHA)
Let pm be a power of a prime, and let σ(pm) denote the sum of divisors of pm. Integer solutions (p, z, m, n) of the equation
(1) σ(pm) = zn, z > 1, m > 1, n > 1 ,
were investigated in many papers. By Nagell [6], (p, z, m, n) = (7, 20, 3, 2) is the only solution of equation (1) with 2 - m. Takaku [8] proved that if (p, z, m, n) is a solution with 2 | n, then p < 22m+1. Chidambaraswawy and Krishnaiah [1] improved this result to p < 22m. However, Ljunggren [4]
and Rotkiewicz [7] showed that the only solutions (p, z, m, n) with 2 | n are (3, 11, 4, 2) and (7, 20, 3, 2). Recently, it was proved by Takaku [9] that if (p, z, m, n) is a solution of (1) such that
(2) m + 1 = qrm1, q - r , q - m1, q | n , q is an odd prime, then p < mq2(2q)(m−1)qm. In this note we prove the following result.
Theorem. Equation (1) has no solution (p, z, m, n) which satisfies (2) with q ≡ 3 (mod 4) .
The proof depends on the next two lemmas, which follow immediately from some old results of Gauss [2; Section 357] and Lucas [5] respectively.
Lemma 1. Let q be an odd prime with q ≡ 3 (mod 4), and let x, y be coprime integers. If q > 3, then
xq− yq
x − y = (A(x, y))2+ q(B(x, y))2,
where A(x, y), B(x, y) are coprime integers with 2A(x, y) ≡ 0 (mod x − y) and 2B(x, y) ≡ 0 (mod xy(x + y)) .
Lemma 2. Let D be a non-square integer , and let x, y be coprime inte- gers. Further , let ε = x + y√
D, ε = x − y√
D, and let E(t) = εt+ εt
ε + ε , F (t) = εt− εt ε − ε
194 M. L E
for any positive integer t with 2 - t. Then E(t), F (t) are integers. Moreover , if E(q)F (q) ≡ 0 (mod p) for some odd primes p, q, then either p = q or p ≡ (D/p) (mod q), where (D/p) is the Legendre symbol.
P r o o f o f T h e o r e m. (1) can be written as
(3) pm+1− 1
p − 1 = zn, z > 1, m > 1, n > 1 .
Let (p, z, m, n) be an integer solution of (3) satisfying (2). By Lemma 4 of [3], this is impossible for q = 3. Below we assume that q > 3.
If p = q, then q | n implies p2| zn−1 (since p | zn−1). So (3) is impossible in this case.
If p 6= q and pm1 6≡ 1 (mod q), then from (3) we get pqr−1m1− 1
p − 1 = z1q and
(4) pm+1− 1
pqr−1m1− 1 = pqr−1m1(q−1)+ . . . + pqr−1m1+ 1 = z2q,
where z1, z2 are positive integers satisfying z1z2 = zn/q. Since p 6≡ 1 (mod q), we have p - (z2q− 1)/(z2− 1) and pqr−1m1| z2− 1 by (4). It fol- lows that
pm+1− 1 = pqrm1− 1 > z2q≥ (pqr−1m1+ 1)q > pqrm1, a contradiction.
If p 6= q, pm1 ≡ 1 (mod q) and q ≡ 3 (mod 4), then q -r implies r = sq−l where s, l are positive integers with l < q. From (3) we get
(5) pm1− 1
p − 1 = qlz0q, pqim1− 1
pqi−1m1− 1 = qziq, i = 1, . . . , r,
where z0, z1, . . . , zr are positive integers satisfying qsz0z1. . . zr = zn/q, 2 - z0z1. . . zr and q - z1. . . zr. We see from (5) that p 6≡ ±1 (mod q). Since r ≥ 1, by Lemma 1 we have
(6) pqm1− 1
pm1− 1 = (A(pm1, 1))2+ q(B(pm1, 1))2= qz1q, where A(pm1, 1), B(pm1, 1) are coprime integers satisfying (7) 2A(pm1, 1) ≡ 0 (mod pm1− 1),
2B(pm1, 1) ≡ 0 (mod pm1(pm1+ 1)) . Hence
(B(pm1, 1))2+ q A(pm1, 1) q
2
= zq1,
PRIMES pWITHσ(pm) = zn 195
where B(pm1, 1), A(pm1, 1)/q are coprime integers. Since the class num- ber of Q(√
−q ) is less than q, it is prime to q. Therefore B(pm1, 1) + (A(pm1, 1)/q)√
−q is the qth power of an algebraic integer of Q(√
−q ). Re- calling that q > 3, we have
(8) B(pm1, 1) + A(pm1, 1) q
√−q = (X1+ Y1
√−q )q,
where X1, Y1 are coprime integers satisfying
(9) X12+ qY12= z1.
Let ε = X1+ Y1
√−q, ε = X1− Y1√
−q. From (7) and (9) we get (10) B(pm1, 1) = X1
εq+ εq ε + ε
≡ 0 (mod pm1) .
Recalling that p 6≡ ±1 (mod q), by Lemma 2 we see from (10) that p - (εq + εq)/(ε + ε) and pm1| X1. If X1 = 0, then gcd(X1, Y1) = 1 shows that Y1 = ±1 and z1 = q by (9), which is impossible. Hence X1 6= 0 and
|X1| ≥ pm1. From (6) and (9) we get
pqm1 > qzq1> X12q≥ p2qm1, a contradiction. Thus the theorem is proved.
Acknowledgment. The author would like to thank the referee for his valuable suggestions.
REFERENCES
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[6] T. N a g e l l, Sur l’´equation ind´etermin´ee (xn− 1)/(x − 1) = y2, Norsk Mat. Forenings Skr. (I) No. 3 (1921), 17 pp.
[7] A. R o t k i e w i c z, Note on the diophantine equation 1 + x + x2+ . . . + xn = ym, Elemente Math. 42 (1987), 76.
[8] A. T a k a k u, Prime numbers such that the sums of the divisors of their powers are perfect squares, Colloq. Math. 49 (1984), 117–121.
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[9] A. T a k a k u, Prime numbers such that the sums of the divisors of their powers are perfect power numbers, Colloq. Math. 52 (1987), 319–323.
RESEARCH DEPARTMENT CHANGSHA RAILWAY INSTITUTE CHANGSHA, HUNAN, CHINA
Re¸cu par la R´edaction le 14.3.1990
