UDC 511

MATHEMATICS

V. V. KOBELEV

PROOF OF FERMAT’S GREAT THEOREM FOR ALL PRIME EXPONENTS LESS THAN 5500

(Presented by Academician S. A. Lebedev on 23 VI 1969)

Kummer (¹) showed that Fermat’s great theorem (FGT) is true for regular prime exponents. In Vandiver’s work (²) a criterion is indicated that makes it possible to decide the question of the validity of FGT for irregular prime exponents. According to Vandiver’s criterion (²), FGT is true for a given irregular prime \(L\), if for all \(Q(a)\)

\[ Q^k \not\equiv 1(\bmod p), \tag{1} \]

where \(p\) is any prime number less than \(L^2 - L\) and of the form \(1 + kL\),

\[ Q = t^{-kd/2} \prod_{j=1}^{\mu} (t^{kj} - 1)^{j^{L-1-2a}}, \tag{2} \]

\(t\) is any integer such that \(t^k \not\equiv 1(\bmod p)\), \(\mu = (L - 1)/2\),

\[ d = \sum_{j=1}^{\mu} j^{L-2a} \tag{3} \]

and \(a < \mu\) are the indices of Bernoulli numbers whose numerators are divisible by \(L\). If, for at least one \(a\) and all admissible \(p\) and \(t\), it turns out that \(Q^k \equiv 1(\bmod p)\), the question of the validity of FGT for the given \(L\) remains open.

Vandiver and collaborators (²–⁴) checked expression (1) on the SWAC computer with an average speed of about 16,000 operations per second (⁵). It turned out that for all irregular primes less than 4002, inequality (1) holds, and thus the validity of FGT was proved for all odd prime numbers \(L < 4002\); moreover, the examination of primes in the range from 2500 to 4000 required about 100 hours of SWAC machine time.

The appearance in recent years of more powerful digital computers makes it possible to check the validity of FGT for a larger range of exponents in an acceptable time. Thus, the use of the BESM-6 computer, performing about one million operations per second (⁶), made it possible to verify the results of works (²–⁴) and to examine primes in the range from 4000 to 5500. We note that examining the primes in the range from 2500 to 4000 took only 31 minutes of BESM-6 machine time.

Checking the results of works (²–⁴) revealed a number of errors in them. The prime numbers 1381, 1597, and 1877 turned out to be irregular, and the degree of irregularity of the prime 1663 turned out to be equal to 2. The table of irregularity of primes with the indicated errors was included in the monograph by Borevich and Shafarevich (⁷).

Table 1 presents the results of computations on BESM-6. The first 6 rows of this table fill the gaps and errors of works (², ⁴). Regular prime numbers are not shown in Table 1.

Table 1

Values of Vandiver’s criterion for irregular prime numbers lying in the range from 4002 to 5500

Since, as is evident from Table 1, $Q^k \not\equiv 1 \pmod p$ for all the irregular $L$ considered, Fermat’s Last Theorem is valid for all odd prime exponents less than 5500. In all cases, for the proof it proved sufficient to use $t = 2$ and the minimal $p$.

Institute of Precision Mechanics and Computer Engineering
Academy of Sciences of the USSR
Moscow

Received
16 VI 1969

CITED LITERATURE

1. E. E. Kummer, J. reine u. angew. Math., 40, No. 2, 130 (1850).
2. D. H. Lehmer, E. Lehmer, H. S. Vandiver, Proc. Nat. Acad. Sci. U.S.A., 40, No. 1, 25 (1954).
3. H. S. Vandiver, Proc. Nat. Acad. Sci. U.S.A., 40, No. 8, 732 (1954).
4. J. L. Selfridge, C. A. Nicol, H. S. Vandiver, Proc. Nat. Acad. Sci. U.S.A., 41, No. 11, 970 (1955).
5. H. D. Huskey, R. Thoresen, B. F. Ambrosio, E. C. Yowell, Proc. IRE, 41, No. 10, 1294 (1953).
6. BESM-6, Basic Technical Data, 1964.
7. Z. I. Borevich, I. R. Shafarevich, Number Theory, Moscow, 1964.