Antal Járai

Largest known twin primes and Sophie Germain primes

The numbers 242206083.2 33880 ±1 are twin primes. The number p = 2375063906985.2 19380 - 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515.2 5056 - 1, the numbers p, p + 2 and 2p + 1 are all primes.

Largest known twin primes

The numbers 697053813 · 216352 ± 1 are twin primes. A search for large prime pairs of the form (3 + 30h) · 2 ± 1 was performed at the University of Paderborn, Germany in the framework of a project for parallel computing in computational number theory supported by the Heinz Nixdorf Institute, Paderborn. We used the arithmetic routines developed in this project for fast parallel and sequential computations with very large numbers. The search was done until the first large prime pair 697053813 · 2 ± 1 was found. These numbers have 4932 digits starting with 1930880535 . . . . To our best knowledge the earlier records were 1692923232·10±1 (4030 digit), 4655478828·10±1 (3439 digit) found by Dubner [1] and 1706595 · 2 ± 1 (3389 digit) found by Parady, Smith, Zarantonello [3] in 1990. The search consisted of the following steps: 1. We started with the 2 nonnegative values of h below 2. The exponent 16352 was fixed during the calculations. 2. Both the case +1 and the case −1 were sieved with factors from 7 up to 2. After sieving, 209571 candidates remained. 3. Using the probabilistic Miller–Rabin primality test (see Knuth [2, pp. 379– 380]), the candidates were tested, first the +1 case and then the −1 case, until a “probable twin prime pair” was found. 4. This “probable twin prime pair” was tested with exact tests: the −1 case using a Lucasian test (see Riesel [5]) and the +1 case using the test of Brillhart, Lehmer and Selfridge and the test of Proth (see Ribenboim [4, pp. 37–40]). The number π2(x) of twin primes below x is conjectured to be π2(x) ∼ 2C2x/ ln (x) where C2 = ∏ p>2(1 − 1/(p − 1)) ≈ 0.66016 . . . . Computer experiments support this conjecture: see Ribenboim [4, p. 202]. Here we test a sequence of numbers N±1, N = (h0 +30h) ·2, where n is a large but fixed natural number, h runs through the positive integers smaller than 2/30, and where h0 ≡ 3 mod 30, n ≡ 0 mod 4 or h0 ≡ 9 mod 30, n ≡ 1 mod 4, or h0 ≡ 21 mod 30, n ≡ 3 mod 4 or h0 ≡ 27 mod 30, n ≡ 2 mod 4 is satisfied, because only in these cases is it true that N − 1 and N + 1 both are not divisible by 2, 3 and 5 and a simple exact Lucas test works for N − 1. Our experiments with a simple Mapler program show that 11/ lnN is a good lower estimate for the probability that N − 1 and N + 1 both are primes for the range N ≤ 2, where n is fixed and h runs up to 2/30. If we assume that this approximation holds up to N ≈ 2, we get ≈ 1/11 000 000 for Received by the editor December 16, 1994. 1991 Mathematics Subject Classification. Primary 11-04; Secondary 11A41. rMaple is a registered trademark of Waterloo Maple Software. c ©1996 American Mathematical Society

On Lipschitz property of solutions of functional equations

SummaryIn this work it is proved that under certain conditions the continuous vector-valued solutionsf of the functional equation

On the Characterization of Weierstrass’s Sigma Function

f(t) = h(t,y,f(g_1 (t,y)), \ldots ,f(g_n (t,y)))

Regularity of functional equations on manifolds

are locally Lipschitz functions. A similar statement is proved for the real-valued solutions of locally bounded variation under weaker additional conditions.

Regularity properties of functional equations in several variables

Using the so-called “almost” variant of a well-known generalization of Steinhaus’ theorem, first we prove a general result on the multiplicative type functional equation (3), then we solve functional equations (1) and (2) originated from statistics under such conditions.

On regular solutions of functional equations

It will be proved that measurable and not almost everywhere zero functions f 1,f 2 : ℝn→ℂ satisfying the functional equation