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.

A final proof the “March on Canberra” is a quarter-century and counting overdue

According to a new paper in the journal Nature, regardless of what the rest of the world does, record-breaking temperature rises are already inevitable until 2040.

At the same time, the Sydney Morning Herald is noting a zero-emissions plan for Britain – whose parity emissions per capita are minimally four times those of Australia – as Australia approves the polluting Adani coal mine. There is – and was even before last month’s surprise election – a certainty Australia will expand fossil fuels whilst the EU moves to zero-net-emissions.

Many (including my brother) naïvely believe that Australia will eventually be condemned as a pariah state for expanding fossil fuels. Nevertheless, this viewpoint overlooks demographic reality. Australia already has substantially higher total fertility than those nations most advanced in decarbonisation. Recent trends towards lowest-low fertility in Finland (from 1.9 to 1.5 children since 2010) and other European nations whose fertility was the least low during the 2000s suggests that Morrison’s policies will widen this gap.

The fact is that – as I have emphasised for two decades – Australia must ecologically have by far the lowest emissions per capita in the world. This demand places human energy consumption upon its natural biological “footing”. Environment, Capitalism and Socialism demonstrated three decades ago that the money existed to finance a rapid transition to a carbon-free Australia as early as 2005 or 2010 – were major polluters taxed severely enough.

As Dimitri Lafleur has partially shown, a carbon-free Australia would remake the world economy by:

1. radically limiting energy and materials use on a global scale, especially in desert nations with naturally low-energy ecologies and zero hydropower potential
2. shifting “developing” economies towards renewable energy once they do not have cheap fossil fuels from Australia and the oil states
3. shifting energy-intensive industries towards those (Enriched and Tropical) nations with large resources in hydropower
4. shifting agriculture towards the high-latitude nations with youngest and most fertile soils
1. this would occur because land clearing is a major source (around 20 percent) of greenhouse emissions in Australia
2. also, Australian soils are thirty thousand times older and more weathered than soils of most other Quaternary landmasses
3. young, high-latitude areas are also least affected by runaway climate change shown as certain by Nature
5. shifting away from planned obsolescence towards long-lasting consumer goods that use fewer resources over the long term

What needed to be done back in the 1990s was for the globe to recognise that – regardless of its relatively small aggregate emissions that have led even environmentalists to neglect it – a rigid, zero-compromise, zero-emissions target for Australia no later than 2010 would have:

1. largely solved global greenhouse gas emissions by radically altering global development patterns
2. paid for the ecological crisis out of the pockets of those people – alongside the Arab Gulf royal families – with greatest duty and ability to pay
3. achieved this in a manner in agreement with Earth’s natural ecology (smallest per-capita energy consumption and emissions in arid desert nations)
4. in an Enriched World then and now crippled by excessive environmental regulations, which stand likely to achieve negligible global gains while Australia mines and uses more and more coal, created major economic opportunities including:
• phase-out of economically crippling Enriched World farm subsidies as Australia’s unsustainable pastoral and broad-acre farmland would be converted to ecotourism
• revitalising such industries as aluminum and titanium smelting when coal use in phased out in Australia and other nations lacking hydropower potential

Given the experiences of the past quarter-century and especially last month’s election, there exists zero possibility that Australia will ever elect a more environmentally responsive government. Thus, other countries are burdened with the critical task of clamping down on the worst environmental performer – a task entirely ignored but unless achieved even a total carbon phase-out in the EU and East Asia will achieve little in the long term.

Instead of decentralised global environmental protests (as seen in recent weeks) what was needed in 1994 and stands three decades overdue was and is a global focus upon the centre of power in the worst environmental performer: protests demanding uncompromising, rapid decarbonisation of Australia, or a “March on Canberra”. Whilst the effects would not be immediate, would be costly to the rest of the world and would need to be sustained over years and even decades, they possess potential to actually deal with the planet’s worst polluter rather than permit Australia indefinite emissions increases negating large-scale decarbonisation abroad.