Euler, Leonard

Petersburg in 1725. Catherine was so desperate to secure Daniel that she agreed to offer a second chair to his brother, Nikolas. Unfortunately, Nikolas died of tuberculosis shortly after arriving in Russia. Despondent over his death, Daniel thought of returning home, but stayed when his father suggested that one of his own students, Leonard Euler, would make an able assistant. 

Leonard Euler (1707-1783), a mathematician bom in Switzerland but who worked in Germany and Russia, proved that every even perfect number is of the form (2p -1) where p is a prime and 2p -1 is also a prime, called a Mersenne prime in honor of Mersenne (1588-1648), a Franciscan Mar who often served as an intermediary in the correspondence between the most prominent mathematicians of his time. For example, if p = 2, a prime, then 2p = 2 -1 = 4-1 = 3 is also a prime and 2 P" 2P -1) = 2 X 3 = 6 which, as we have seen is indeed a perfect number. If p = 3, the next prime, then 2p -1=2 -1 = 8-1 = 7 is again a prime. This gives us 2 X 7 = 4 X 7 = 28 which, as we have seen, is also a perfect number. The first case where p is a prime but 2p -1 is not occurs when p = 11. Here we have 2 -1 = 2048-1 = 23 X 89. 

Leonard, B. P. (1988), Third-order multidimensional monotonic Euler/ Navier Stokes Solver, Draft for First National Fluid Dynamic Conference, Cincinnati, Ohio, July 1988. 

The Euler s formula developed by Leonard Euler (Swiss mathematician, 1707 to 1783) is used in product designs and also in designs using columns in molds and dies that process plastic. Euler s formula assumes that the failure of a column is due solely to the stresses induced by sidewise bending. This assumption is not true for short columns that fidl mainly by direct compression, nor is it true for columns of medium length. The failure in such cases is by a combination of direct compression and bending. 

The above equation is known as the Euler—Lagrange equation in honor of Swiss mathematician Leonard Euler (1707-1783) and French mathematician Joseph Luis de Lagrange (1736-1813). The Euler-Lagrange equation is also called the adjoint or costate equation since it defines the adjoint or costate variable A. 

The first concrete public key system proposed with a security that has withstood the test of time was the RSA system, named for its inventors Ronald Rivest, Adi Shamir, and Leonard Adleman. RSA is based on the computational difficulty of factoring the product of large primes. The public key consists of an n-bit modulus N, which is the product of two primes p and q each of length n/2 bits, and an element e of the multiplicative group Z. /V is called the RSA modulus and e is called the encryption exponent. The decryption exponent is d such that ed = mod (pin), where (piN) = ip — )iq — 1) is the Euler totient function. The private key is the pair N, d). Once the public and private keys have been generated, p and q are no longer needed. They can be discarded, but should not be disclosed. 

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