Golden Ratio of Primes (Amateur)

Question

Unable to find information elsewhere, so I'll try here.

What two consecutive primes are closest to producing the Golden Ratio? Or two of any Primes?

Has this been determined?

Thanks!

Answer 1

Of any two primes...? Well, for each prime $p$, let's compute the minimum value of $d = \|\log{\phi p} - \log{q}\|$ over all primes $q$. Then pairs $(p,q)$ with a ratio close to $\phi$ will have a small value of $d$. Here's a plot of $\log d$ versus $p$ for $p<10^4$:

Good pairs stand out as downward spikes on the graph. Some notable pairs are $(29,47)$, $(97,157)$, $(563,911)$, $(631,1021)$, $(1453,2351)$ and the best pair in this range: $(2207,3571)$. For that last pair, we have $2207\phi = 3571.00101\ldots$, which is pretty impressive!

For fun, here's the raw data for $p<100$, formatted as $p, q, d, \log d$:

3, 5, 0.0296138, -3.51951 5, 7, 0.14474, -1.93282 7, 11, 0.0292267, -3.53267 11, 17, 0.0458938, -3.08143 13, 23, 0.089333, -2.41538 17, 29, 0.0528707, -2.93991 19, 31, 0.0083364, -4.78712 23, 37, 0.00578813, -5.15195 29, 47, 0.00163995, -6.41309 31, 53, 0.0550929, -2.89873 37, 59, 0.0145923, -4.22726 41, 67, 0.00990873, -4.61434 43, 71, 0.0202679, -3.89872 47, 79, 0.0380884, -3.26784 53, 83, 0.0326631, -3.42151 59, 97, 0.0159617, -4.13756 61, 97, 0.0173747, -4.05274 67, 109, 0.00544344, -5.21334 71, 113, 0.0165039, -4.10416 73, 113, 0.0442834, -3.11714 79, 127, 0.00647259, -5.04018 83, 137, 0.0199285, -3.9156 89, 149, 0.0340981, -3.37851 97, 157, 0.000323002, -8.03785