Rarity of full-period prime denominators in rational approximations of irrational numbers

Ever since, a few months ago, I began to discover that – in addition to the very well-known approximation of 22/7 for 𝜋 – other irrational numbers had frequently used rational approximations, I have been struck by one fact: that full-period primes occur very rarely as denominators in such fractions, especially in the most useful rational approximations for hand calculations.

In order to test this hypothesis I have compiled a representative selection of irrational numbers in the table below, useful rational approximations for these numbers, and the periods of these rational approximations. For most numbers the most common rational approximation has been used; in certain cases like the square root of 6 and 𝜋, I have given more than one rational approximation, with that with the larger denominator naturally more accurate.

Number	Decimal expansion	Rational Approximation	Period	Prime factorisation of denominator	Character and type of prime factors
√2	1.4142135623730950488016887242	99/70	6	2•5•7	Composite Full-period and terminating factors
√3	1.7320508075688772935274463415	97/56	7	2•2•2•7	Composite Full-period and terminating factors
√5	2.2360679774997896964091736687	161/72	1	2•2•2•3•3	Composite Short-period (unique) and terminating factors
√6	2.4494897427831780981972840747	49/20	0	2•2•5	Composite Terminating decimal
		218/89	44	89	Half-period prime
√7	2.6457513110645905905016157536	127/48	1	2•2•2•2•3	Composite Short-period (unique) and terminating factors
√10	3.1622776601683793319988935444	117/37	3	37	Short-period (unique) prime
√11	3.3166247903553998491149327366	199/60	1	2•2•3•5	Composite Short-period (unique) and terminating factors
∛2	1.2599210498948731647672106072	63/50	0	2•5•5	Composite Terminating decimal
∛3	1.4422495703074083823216383107	75/52	6	2•2•13	Composite Half-period prime factor
∛4	1.5874010519681994747517056392	100/63	6	3•3•7	Composite Half-period and short-period (unique) factors
		227/143	6	11•13
∛5	1.7099759466766969893531088725	171/100	0	2•2•5•5	Composite Terminating decimal
∛6	1.8171205928321396588912117563	467/257	256	257	Full-period prime (accurate to 1-in-33,629,323!)
∜2	1.1892071150027210667174999705	44/37	3	37	Short-period (unique) prime
∜3	1.3160740129524924608192189017	25/19	18	19	Full-period prime
		229/174	28	2•3•29	Composite Full-period, short-period and terminating factors
21/5	1.1486983549970350067986269467	85/74	3	2•37	Composite Short-period (unique) and terminating factors
21/12	1.0594630943592952645618252949	89/84	6	2•2•3•7	Composite Full-period, short-period and terminating factors
𝜋	3.1415926535897932384626433832	22/7	6	7	Full-period prime
		355/113	112	113	Full-period prime
e	2.7182818284590452353602874713	193/71	35	71	Half-period prime
ee	15.154262241479264189760430272	197/13	6	13	Half-period prime
		2849/188	46	2•2•47	Composite Full-period prime factor
e𝜋	23.140692632779269005729086367	1481/64	0	2•2•2•2•2•2	Terminating decimal
ln 2	0.6931471805599453094172321214	61/88	2	2•2•2•11	Composite Short-period (unique) prime factor
log10 2	0.3010299956639811952137388947	59/196	42	2•2•7•7	Composite Terminating and squared full-period prime factor

If we study this table, we see that, for whatever reason, there seem to be very few full-period prime denominators. The two well-known approximations for 𝜋, one approximation for the fourth root of three, and one remarkable approximation for the cube root of six are the only exceptions. [The cube root of six – which has minor notability as the geometric mean of 1, 2 and 3 – I did not originally intend to include but decided to do so because the approximation ⁴⁶⁷/₂₅₇ is so amazingly accurate, being superior to the famous Milü approximation for 𝜋].

Why this should be so is an interesting question. It is possibly because the way in which the continued fractions used to find such approximations as ⁴⁶⁷/₂₅₇ for ∛6 would add factors in the finding of “common denominators” needed for addition of fractions, although I have not checked this yet.