The Wikipedia page suggest that in the epic, Gilgamesh is a demigod of superhuman strength.
How do we know he was a demigod? What makes someone a demigod?
Did he possess some special powers that make him a superhuman?
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In general, English "demigod" refers to a person who is partially divine and partially human. Most typically, this refers to someone who has one divine and one human parent (like Hercules, or the Pandavas), making them 50% divine. But the case of Gilgamesh is slightly different. Rather than being half-divine, Gilgamesh is in fact two-thirds divine.
When the gods created Gilgamesh they gave him a perfect body. Shamash the glorious sun endowed him with beauty, Adad the god of the storm endowed him with courage, the great gods made his beauty perfect, surpassing all others, terrifying like a great wild bull. Two thirds they made him god and one third man.
From a mathematical perspective, you can only be two-thirds something if 1.) you have a very finely-crafted family tree extending infinitely far back [i.e. successive binary approximations of 2/3]; or 2.) you have loops in your family tree [which requires time travel, probably]. (This is because 2/3 is not a binary fraction).
So how is it that Gilgamesh was two-thirds divine if born from a divine mother (Ninsun) and a mortal father (Lugalbanda)? I don't know, but there you have it. I can't imagine mathematical accuracy would have been a priority of the Sumerians when they crafted the Epic.
The definition for demigod differs from culture to culture. But the definition of demigod, which can be used here in the case of Gilgamesh would be "half-god".
Gilgamesh was a Sumerian king who wished to become immortal. Endowed with superhuman strength, courage, and power, he appeared in numerous legends and myths, including the Epic of Gilgamesh.
From Myth Encyclopedia article on Gilgamesh (emphasis mine):
According to legendary accounts, Gilgamesh was the son of the goddess Ninsun and of either Lugalbanda, a king of Uruk, or of a high priest of the district of Kullab.
The fact that his mother was a goddess and his father, a mortal, suggests than Gilgamesh was a demigod. And his superhuman abilities could have been due the fact that one of his parents was a god. In the Epic, his strength and feats were described as superhuman.