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26. Mar 2023•0 gefällt mir•2 views

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Show Goldbach?s conjecture that every even integer greater than 2 is the sum of two primes is equivalent to the conjecture that every integer greater than 5 is the sum of three primes (Hint: If is n > 5, consider n ? 2 if n is even and n - 3 if n is odd, then use Goldbach; Conversely, if n > 2 is even, consider n + 2.) Solution Every even integer greater than 2 is the sum of two primes. Consider any integer >5. Let this number be l. Either l can be odd or l can be even. If l is odd, l-3 is even>2 and can be expressed as sum of two primes by the Goldbach. Thus l-3 = Sum of two primes Hence l = sum of two primes + 3 = sum of 3 primes --------------------------------------------------------------------------------- If l is even, l-2>2 Hence l-2 can be expressed as sum of two primes As a result l = 2+sum of 2 primes = sum of 3 primes. Tus proved.

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- 1. Solve the following graphically Max z = 3x1 +4x2 s.t. x1 + 2x2 = 16 2x1 + 3x2 = 18 x1 = 2 x2 = 10 x1, x2 = 0 What are the optimal values of x1, x2, and z? Write your answer in the form : ( x1, x2, z). (For example, the expression (10, 20, 50) means that x1 = 10, x2 = 20, and z = 50). Solution From graphical method, (9,0,27) are the optimal values of x1, x2, and z. (x1 = 9, x2 = 0, z = 27)