Abstract. We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.

Abstract. Using the properties of elliptic curves, we propose primality tests for integers of the form 2 p ± 2 (p+1)/2 + 1, where p is a prime number.

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Let K be a finite field. We know that half of the elements of K * are square. So it is natural to ask how many of them appear as x-coordinates of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite fields and show that half of the x-coordinates on an elliptic curve are square.