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.

Table1 5n+1 vs 5n+3

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).

Table 2b 5n+1 vs 5n+3

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.

Table 3b 5n+1 vs 5n+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.

Table 4b 5n+1 vs 5n+3

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

Table 5a 5n+1 vs 5n+3 Table 5b 5n+1 vs 5n+3

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?

*

Bandung, 05-09-2016

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