The 5n + 1 vs 5n + 3 Problem*

(Artikel ini sengaja saya tulis dalam bahasa Inggris, supaya dapat dibaca pula oleh mereka yang tidak bisa berbahasa Indonesia.)

We have discussed the 5n + 1 problem which has similar behavior to the 3n + 1 problem. Out of curiosity, I investigated what happens if we do 5n + 3 instead of 5n + 1, using a computer program made by Mr. Bee.

It turns out that some sequences ‘converge’ to 1 (as in the 5n + 1 problem), but some escape to infinity and the others are trapped in a loop. For examples, for n₁ = 25, the sequence converges to 1; for n₁ = 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 47, or 49 (and many others), the sequences run off to infinity; and for n₁ = 43, 53, or 61 (and many others), the sequences are trapped in a loop. Below is the sequence obtained for n₁ = 43.

My curiosity went further. Knowing the behavior of the sequences in the 5n + 3 problem, I would like to find out what happens if we do 5n + 1 and 5n + 3 alternately, every time we hit a number which is not divisible by 2 or 3. For example, for n₁ = 5, the sequence converges to 1. The same also occurs for n₁ = 43. However, if we start with n₁ = 23, then we end up with a loop, as in the following table (and the next one).

Note that at Step 17, when we get 11, we do 5n + 3 and get n₁₈ = 58 and n₁₉ = 29. Next, we do 5n + 1 and get the following sequence (just add the step by 18), where we eventually go back to the previous Step 3.

It is interesting to observe what happens if we change the order of 5n + 1 and 5n +3 operations. For example, if we start with n₁ = 11, with 5n + 1 operation first, then we will get a sequence which is convergent to 1; but if we apply 5n + 3 operation first, then we will get a loop.

Some loops are long, for example for n₁ = 179, with 5n + 3 operation first.

 

The 5n + 1 vs 5n + 3 problem seems to have interesting behavior to investigate further. One question really bothers me: is there a sequence that runs off to infinity? If not, then the 5n + 3 operations seem to never win against the 5n + 1 operations; they always lose to or tie with the 5n + 1 operations. Is it really the case?

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Bandung, 05-09-2016