Problem 57535. Find patterns in subprime Fibonacci sequences

Created by ChrisR

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 321px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 160.5px; transform-origin: 407px 160.5px; vertical-align: baseline; "><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 373.042px 8px; transform-origin: 373.042px 8px; unicode-bidi: normal; "><span style="">Lots of Cody problems involve Fibonacci and Collatz sequences. Fibonacci sequences start with two numbers, and later terms are computed by summing the previous two terms. The terms continue to increase, of course. Terms in Collatz sequences are computed with </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/21"><span style=""><span style="text-decoration-line: underline; ">a different formula</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226px 8px; transform-origin: 226px 8px; unicode-bidi: normal; "><span style="">, and for the initial values that have been tried, the sequences eventually reach a 1. An unsolved problem is whether Collatz sequences reach 1 for any initial value. </span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="">The problem combines ideas from these two sequences by examining subprime Fibonacci sequences. The sequence starts with two given values, and the next term is computed as the sum of the previous two. However, if the sum is composite, it is divided by the smallest prime factor. If the starting values are [1 1], then the sequence is 1, 1, 2, 3, 5, 4, 3, 7, 5, etc. </span></span></div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.758px 8px; transform-origin: 375.758px 8px; unicode-bidi: normal; "><span style="">Just as all Collatz sequences reach 1 (so far), all subprime Fibonacci sequences reach a repeating pattern. However, the pattern changes with the starting values. With starting values [1 1], the repeating pattern is 18 terms long, and it starts on term 38. With starting values [7 37], the repeating pattern is 136 terms long, and it starts on term 37. The subprime Fibonacci conjecture, from </span></span><a target='_blank' href = "https://arxiv.org/pdf/1207.5099.pdf"><span style=""><span style="text-decoration-line: underline; ">this paper</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 187.475px 8px; transform-origin: 187.475px 8px; unicode-bidi: normal; "><span style="">, is that a repeating pattern is reached for all starting values. </span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 371.7px 8px; transform-origin: 371.7px 8px; unicode-bidi: normal; "><span style="">Write a function that takes a vector of two starting values and produces the repeating pattern, the number of the starting term, and the length of the pattern. Can you find patterns of lengths other than those in the test suite? Can you prove the subprime Fibonacci conjecture? </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "><span style=""> </span></span></div></div></div>

Lots of Cody problems involve Fibonacci and Collatz sequences. Fibonacci sequences start with two numbers, and later terms are computed by summing the previous two terms. The terms continue to increase, of course. Terms in Collatz sequences are computed with a different formula, and for the initial values that have been tried, the sequences eventually reach a 1. An unsolved problem is whether Collatz sequences reach 1 for any initial value. 
The problem combines ideas from these two sequences by examining subprime Fibonacci sequences. The sequence starts with two given values, and the next term is computed as the sum of the previous two. However, if the sum is composite, it is divided by the smallest prime factor. If the starting values are [1 1], then the sequence is 1, 1, 2, 3, 5, 4, 3, 7, 5, etc. 
Just as all Collatz sequences reach 1 (so far), all subprime Fibonacci sequences reach a repeating pattern. However, the pattern changes with the starting values. With starting values [1 1], the repeating pattern is 18 terms long, and it starts on term 38. With starting values [7 37], the repeating pattern is 136 terms long, and it starts on term 37. The subprime Fibonacci conjecture, from this paper, is that a repeating pattern is reached for all starting values. 
Write a function that takes a vector of two starting values and produces the repeating pattern, the number of the starting term, and the length of the pattern. Can you find patterns of lengths other than those in the test suite? Can you prove the subprime Fibonacci conjecture?

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Last Solution submitted on Feb 04, 2023

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William on 12 Jan 2023

All the tests except #2 have a curious notation for the value of q_correct, and it is undefined.

Ramon Villamangca on 12 Jan 2023

Unrecognized function or variable 'q18'.

ChrisR on 12 Jan 2023

Sorry. It should work now.

I tried defining the patterns at the top of the code, and I should have known that Cody would strip that code away. It happened in just about every problem in the MATLAB Fundamentals - Plotting and Visualization group.

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Problem 57535. Find patterns in subprime Fibonacci sequences

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