1.Solve the equation 3 ∗ x ≡ b (mod 7), for the unknown variable x . Conside

Question

1.Solve the equation 3∗x ≡ b (mod 7), for the unknown variable x. Consider all possible values for b, i.e., b is a natural number that takes distinct values from 0 to 6. (So there are seven separate equations to solve here, and you will get seven solutions. Your answers should be between 0 and 6 inclusive.)

2. Use Fermat’s little theorem to compute: (a) 3200 (mod 19) (b) 18200 (mod 19)

3. (a) Use the Extended Euclidean algorithm to find an inverse of 55 modulo 144. (Your answer should be between 1 and 143.)

(b) Modify the result from the first part of this problem to compute an inverse of 144 modulo 55. (Your answer should be between 1 and 54.)

Explanation / Answer

1

Abstract

Cerium based intermetallic compounds exhibit a wealth of physical properties originating from the electronic states of Ce, i. e. diamagnetic Ce4+ ([Xe] 4f0), paramagnetic Ce3+ ([Xe] 4f1). Switching between the electronic states can be induced either chemically such as by inserting hydrogen, by substitutions (size effects) or physically by applying external pressure. The review exposes different classes of Ce intermetallic compounds whose properties are interpreted and/or predicted thanks to quantum computations in the framework of the density functional theoretical (DFT). Focus is broadly made on the family of the equiatomic cerium intermetallic compounds, namely ternary CeTX where T is a transition metal and X a p-element where the hydrogenation effects take a considerable place in changing the electronic configuration of Ce. Other stoichiometries of cerium intermetallic compounds with their physical properties are discussed subsequently in the later part of the review. Rather than presenting an exhaustive enumeration of stoichiometries, illustrative case studies are selected for each class of materials to provide, after presenting the experimental context, insights into original outcome from methods targeted at selective physical and chemical properties.
Keywords
Cerium intermetallics; hydrogenation; DFT; chemical bonding; magnetism; electronic structure; elastic constants

3

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.

This theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems".

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