(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Question

Suppose we have a linear program in standard equation form maximize cTx subject to Ax = b. x ≥ 0. and suppose u, v, and w are all optimal solutions to this linear program. (a) Prove that zu+v+w is an optimal solution. (b) If you try to adapt your proof from part (a) to prove that that u+v+w is an optimal solution, say exactly which part(s) of the proof go wrong. (c) If you try to adapt your proof from part (a) to prove that u+v-w is an optimal solution, say exactly which part(s) of the proof go wrong.

Answer 1

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Suppose we have a linear program in standard equation form maximize cTx subject to Ax = b. x ≥ 0. and suppose u, v, and w are all optimal solutions to this linear program. (a) Prove that zu+v+w is an optimal solution. (b) If you try to adapt your proof from part (a) to prove that that u+v+w is an optimal solution, say exactly which part(s) of the proof go wrong. (c) If you try to adapt your proof from part (a) to prove that u+v-w is an optimal solution, say exactly which part(s) of the proof go wrong.

a) Suppose that we are carrying out the 1-phase simplex algorithm on a linear program in standard inequality form (with 3 variables and 4 constraints) and suppose that we have reached a point where we have obtained the following tableau. Apply one more pivot operation, indicating the highlighted row and column and the row operations you carry out. What can you conclude from your updated tableau? x1 x2 x3 81 82 83 84 81 -2 0 1 1 0 0 0 3 82 3 0 -2 0 1 2 0 6 12 1 1 -3 0 0 1 0 2 84 -3 0 2 0 0 -1 1 4 -2 -2 0 11 0 0-4 0 -8

Microsoft Excel snapshot for random sampling: Also note the formula used for the last column 02 x✓ fx =INDEX(5852:58551, RANK(C2, $C$2:$C$51)) A B 1 No. States 2 1 ALABAMA Rand No. 0.925957526 3 2 ALASKA 0.372999976 4 3 ARIZONA 0.941323044 5 4 ARKANSAS 0.071266381 Random Sample CALIFORNIA NORTH CAROLINA ARKANSAS WASHINGTON G7 Microsoft Excel snapshot for systematic sampling: xfx INDEX(SD52:50551, F7) A B E F G 1 No. States Rand No. Random Sample population 50 2 1 ALABAMA 0.5296685 NEW HAMPSHIRE sample 10 3 2 ALASKA 0.4493186 OKLAHOMA k 5 4 3 ARIZONA 0.707914 KANSAS 5 4 ARKANSAS 0.4831379 NORTH DAKOTA 6 5 CALIFORNIA 0.7277162 INDIANA Random Sample Sample Name 7 6 COLORADO 0.5865002 MISSISSIPPI 8 7:ONNECTICU 0.7640596 ILLINOIS 9 8 DELAWARE 0.5783029 MISSOURI 525 10 15 INDIANA MARYLAND COLORADO

Answer 2

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 3

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 4

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 5

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 6

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Answer 7

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Answer 8

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

Answer 9

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

Answer 10

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

Answer 11

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

Answer 12

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

Answer 13

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

Answer 14

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

Answer 15

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

Answer 16

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 17

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 18

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Answer 29

Ch. 6.1 - Complete the crossword puzzle shown on the...Ch. 6.1 - Prob. 2EE Ch. 6.1 - Write 357 as a traditional Chinese numeral.Ch. 6.1 - Prob. 4EE Ch. 6.1 - Prob. 5EE Ch. 6.1 - Prob. 6EE Ch. 6.1 - Prob. 1ES Ch. 6.1 - Prob. 2ES Ch. 6.1 - Prob. 3ES Ch. 6.1 - Prob. 4ES

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Solution Summary: The author analyzes whether the statement 'There are infinitely many twin primes' is a theorem or conjecture.

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Chapter 6 Solutions

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Solution Summary: The author analyzes whether the statement 'There are infinitely many twin primes' is a theorem or conjecture.

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b) Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c) Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d) Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e) Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f) Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Chapter 6 Solutions

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.