Sidon sequence

In number theory, a Sidon sequence (or Sidon set), named after the Hungarian mathematician Simon Sidon, is a sequence A = {a₀, a₁, a₂, ...} of natural numbers in which all pairwise sums a_i + a_j (i ≤ j) are different. Sidon introduced the concept in his investigations of Fourier series.

The main problem in the study of Sidon sequences, posed by Sidon, is to find the largest number of elements a Sidon sequence A can have smaller than some given number x. Despite a large body of research, the question remained unsolved for almost 80 years. In 2010, it was finally settled by J. Cilleruelo, I. Ruzsa and C. Vinuesa.

Early results

Paul Erdős and Pál Turán proved that, for every x > 0, the number of elements smaller than x in a Sidon sequence is at most . Using a construction of J. Singer, they showed that there exist Sidon sequences that contain terms less than x.

Infinite Sidon sequences

Erdős also showed that if we consider any particular infinite Sidon sequence A and let A(x) denote the number of its elements up to x, then

Sidon

Coordinates: 33°33′38″N 35°23′53″E / 33.56056°N 35.39806°E / 33.56056; 35.39806

Sidon or Saïda (Arabic: صيدا‎, صيدون, Ṣaydā; Phoenician: , Ṣdn; Biblical Hebrew: צִידוֹן, Ṣīḏōn; Greek: Σιδών; Latin: Sidon; Turkish: Sayda) is the third-largest city in Lebanon. It is located in the South Governorate of Lebanon, on the Mediterranean coast, about 40 kilometres (25 miles) north of Tyre and 40 km (25 miles) south of the capital Beirut. In Genesis, Sidon is a son of Canaan, a grandson of Noah. Its name coincides with the modern Arabic word for fishery.

History

Sidon (whose name in classical Arabic is: صَيْدونْ (Saydoon)) has been inhabited since very early in prehistory. The archaeological site of Sidon II shows a lithic assemblage dating to the Acheulean, whilst finds at Sidon III include a Heavy Neolithic assemblage suggested to date just prior to the invention of pottery. It was one of the most important Phoenician cities, and may have been the oldest. From here, and other ports, a great Mediterranean commercial empire was founded. Homer praised the skill of its craftsmen in producing glass, purple dyes, and its women's skill at the art of embroidery. It was also from here that a colonizing party went to found the city of Tyre. Tyre also grew into a great city, and in subsequent years there was competition between the two, each claiming to be the metropolis ('Mother City') of Phoenicia. Glass manufacturing, Sidon's most important enterprise in the Phoenician era, was conducted on a vast scale, and the production of purple dye was almost as important. The small shell of the Murex trunculus was broken in order to extract the pigment that was so rare it became the mark of royalty.

List of Pokémon (102–151)

The Pokémon (ポケモン, Pokemon) franchise has 721 (as of the release of Pokémon Omega Ruby and Alpha Sapphire) distinctive fictional species classified as the titular Pokémon. This is a selected listing of 49 of the Pokémon species, originally found in the Red and Green versions, arranged as they are in the main game series' National Pokédex.

Exeggcute

Exeggcute (タマタマ, Tamatama), known as the Egg Pokémon, is a single Pokémon with six separate seed bodies (often mistaken for eggs, according to the Pokémon episode, "March of the Exeggutor Squad"). Exeggcute's name is a play on the word "execute". Tamatama could be taken to mean 偶然, unexpectedly, although it is more likely to be derived from 玉 tama, sphere, doubled. It is worth noting that the Japanese word for egg is 卵 tamago. They use telepathic signals only they can receive to remain in their balanced formation; even if one is separated, it will quickly rejoin its swarm. Each member of the swarm has a different appearance in number of cracks, deformations, personalities and facial expressions. Some Exeggcute can have such severe cracking and breakage that a yolk can be seen inside. The middle body often has the fewest or no cracks at all, and is assumed to be the boss of the group.

Sidon (surname)

Sidon is the surname of: