Question

What is the period-length of the Linear Congruential Generator Xn+1 = (2Xn + 5)mod13 with seed Xo = 8? Numeric Answer: ! Cannot be empty

Answer 1

(1)

Given,

Linear Congruential Generator X_n+1 = (2X_n + 5) mod 13 , sees X₀ = 8

X₀ = 8

X₁ = (2X₀ + 5) mod 13 = (2*8+ 5) mod 13= 21 mod 13 = 8.

X₂ = (2X₁ + 5) mod 13 = (2*8+ 5)mod 13= 21 mod 13 = 8.

X₃ = (2X₂ + 5) mod 13 = (2*8+ 5)mod 13= 21 mod 13 = 8

Therefore, X_n = {8, 8, 8,...}

Therefore, Sequence is (8) and Period length is 1.

(2)

Given,

Linear Congruential Generator X_n+1 = (2X_n + 5) mod 13 , sees X₀ = 1

X₀ = 1

X₁ = (2X₀ + 5) mod 13 = (2*1+ 5) mod 13= 7 mod 13 = 7.

X₂ = (2X₁ + 5) mod 13 = (2*7+ 5)mod 13= 19 mod 13 = 6.

X₃ = (2X₂ + 5) mod 13 = (2*6+ 5) mod 13= 17 mod 13 = 4.

X₄ = (2X₃ + 5)mod 13 = (2*4+ 5)mod 13 = 13 mod 13 = 0.

X₅ = (2X₄ + 5) mod 13 = (2*0+ 5)mod 13 = 5 mod 13 = 5.

X₆ = (2X₅ + 5)mod 13 = (2*5+ 5)mod 13 = 15 mod 13 = 2.

X₇ = (2X₆ + 5)mod 13 = (2*2+ 5)mod 13 = 9 mod 13 = 9.

X₈ = (2X₇ + 5)mod 13 = (2*9+ 5)mod 13 = 23 mod 13 = 10.

X₉ = (2X₈ + 5)mod 13 = (2*10+ 5)mod 13 = 25 mod 13 = 12.

X₁₀ = (2X₉ + 5)mod 13 = (2*12+ 5)mod 13 = 29 mod 13 = 3.

X₁₁ = (2X₁₀ + 5)mod 13 = (2*3+ 5)mod 13 = 11 mod 13 = 11.

X₁₂ = (2X₁₁ + 5)mod 13 = (2*11+ 5)mod 13 = 27 mod 13 = 1.

X₁₃ = (2X₁₂ + 5)mod 13 = (2*1+ 5)mod 13 = 7 mod 13 = 7.

X14= (2X₁₃ + 5)mod 13 = (2*7+ 5)mod 13 = 19 mod 13 = 6.

X₁₅ = (2X₁₄ + 5)mod 13 = (2*6+ 5)mod 13 = 17 mod 13 = 4.

Therefore, X_n = {1, 7, 6, 4, 0, 5, 2, 9, 10, 12, 3, 11, 1, 7, 6, 4, ....}

Therefore, Sequence is (1-7-6-4-0-5-2-9-10-12-3-11) and Period length is 12.

(3)

Given,

Linear Congruential Generator X_n+1 = (1X_n + 5) mod 13 , sees X₀ = 1

X₀ = 1

X₁ = (1X₀ + 5) mod 13 = (1*1+ 5) mod 13= 6 mod 13 = 6.

X₂ = (1X₁ + 5) mod 13 = (1*6+ 5)mod 13= 11 mod 13 = 11.

X₃ = (1X₂ + 5) mod 13 = (1*11+ 5) mod 13= 16 mod 13 = 3.

X₄ = (1X₃ + 5)mod 13 = (1*3+ 5)mod 13 = 8 mod 13 = 8.

X₅ = (1X₄ + 5) mod 13 = (1*8+ 5)mod 13 = 13 mod 13 = 0.

X₆ = (1X₅ + 5)mod 13 = (1*0+ 5)mod 13 = 5 mod 13 = 5.

X₇ = (1X₆ + 5)mod 13 = (1*5+ 5)mod 13 = 10 mod 13 = 10.

X₈ = (1X₇ + 5)mod 13 = (1*10+ 5)mod 13 = 15 mod 13 = 2.

X₉ = (1X₈ + 5)mod 13 = (1*2+ 5)mod 13 = 7 mod 13 = 7.

X₁₀ = (1X₉ + 5)mod 13 = (1*7+ 5)mod 13 = 12 mod 13 = 12.

X₁₁ = (1X₁₀ + 5)mod 13 = (1*12+ 5)mod 13 = 17 mod 13 = 4.

X₁₂ = (1X₁₁ + 5)mod 13 = (1*4+ 5)mod 13 = 9 mod 13 = 9.

X₁₃ = (1X₁₂ + 5)mod 13 = (1*9+ 5)mod 13 = 14 mod 13 = 1.

X14= (1X₁₃ + 5)mod 13 = (1*1+ 5)mod 13 = 6 mod 13 = 6.

X₁₅ = (1X₁₄ + 5)mod 13 = (1*6+ 5)mod 13 = 11 mod 13 = 11.

Therefore, X_n = {1, 6, 11, 3, 8, 0, 5, 10, 2, 7, 12, 4, 9, 1, 6, .....}

Therefore, Sequence is (1-6-11-3-8-0-5-10-2-7-12-4-9) and Period length is 13.