Ok here is a new hypothesis. Not related to Prime numbers, but cryptography itself. I know it is a long shot. But it is just a thinking exercise.

I must thank my good friend Wyatt for explaining sesquation and quotation to me. Even though I am still trying to learn how to spell them. Yes I didn't make up sesquation or quotation. They are actually a thing in math. I have never encountered them before. So if anyone knows any links describing them please let me know.

I know my hypothesis is off the wall, but I am quite serious. I reiterate that I did not make up sesquation and quotation. This is the first I heard of them. I am only pointing out these simple methods have uses in cryptography and also multivariable equations. Yes I know it sounds like nonsense, but a one-way-function is not just solving an impossible pattern. Instead it can be approached as solving for more than one unknown variable.

Sesquation:

2+3 = 5

2*3 = 6

5+6 = 11

11/2 = 5.5

Quotation:

2+3

4+6 = 10

4*6 = 24

10+24=34

34/4 = 8.5

Patterns? Simple enough but does it describe slope, derivation, cryptography, or Trurl's Method?

Taking the values from the last tabulation of Quotation above:

34-8.5 = 25.5

25-10 = 15.5

______

10+18.5 = 18.5

34-18.5 = 15.5

The Hypothesis is that the methods of sesquation and quotation can solve equations of multiple unknowns, if we know the end values of these methods. It could also be used to solve polynomial equations if we treat the variable in the polynomial equation as separate variables as the variable of the equation is raised to different power.

For example for the polynomial method of the equation: (((pnp^2/ x ) + x^2) / pnp) = 1; replace x^2 with an unknown variable (say z) and plug it into quotation. Yes you have to know multiple answers of the quotation, but in cryptography the user is given many wrong tries compared to one right answer. There are many incorrect answers to work with. The goal is to see how the cypher effects the variables. This means if the user can find a pattern where the values produced randomness, then maybe the message could be solved. Yes I now that these patterns are hidden by cryptographic hash, but I am referring to pattern of the open source equation of the public key cryptography.

1 hour ago, Trurl said:

The Hypothesis is that the methods of sesquation and quotation can solve equations of multiple unknowns, if we know the end values of these methods. It could also be used to solve polynomial equations if we treat the variable in the polynomial equation as separate variables as the variable of the equation is raised to different power.

Well I have heard of 'sesqui...' in mathematics

Sesqui comes form the Latin 'sesqui ' meaning one and a half.

For instance you have sesquilinear forms and antilinear maps.

For two variables on a complex linear space, one can be linear and the other antilinear or sesquilinear.

These devices appear in quantum mechanics amongst other places.

The sesqui prefix is also used in Chemistry to descibe a mixture containing two kinds of radicals in the proportions 2 : 3

But I have never heard of quotation spaces.

Perhaps you mean quotient spaces in higher algebra ?

Edited April 30Apr 30 by studiot

Perhaps @exchemist or @sethoflagos will tell us about the use of sesqui in music ( I have just discovered it is important there) or Chemistry.

3 hours ago, studiot said:

Perhaps @exchemist or @sethoflagos will tell us about the use of sesqui in music ( I have just discovered it is important there) or Chemistry.

I gather in music the Latin for hemiola is sequialtera, both meaning 3:2. But I'm only familiar with hemiola.

Over here "sesqui" came to mean "disaster".

https://en.wikipedia.org/wiki/Sesqui_1990

6 minutes ago, pzkpfw said:

Over here "sesqui" came to mean "disaster".

https://en.wikipedia.org/wiki/Sesqui_1990

Yes the one and a half centenary.

My Oxford English has

sesquicentennial

Also

sesquialtera (music)

sesquioxide (chemistry)

sesquiplane (aero engineering)

sesquiterpene (chemistry)

sesquitertia (maths)

sesquipedal (organ music)

37 minutes ago, exchemist said:

I gather in music the Latin for hemiola is sequialtera, both meaning 3:2. But I'm only familiar with hemiola.

Being a musical dunce I have no idea what a hemiola is.

2 hours ago, exchemist said:

I gather in music the Latin for hemiola is sequialtera, both meaning 3:2. But I'm only familiar with hemiola.

Ditto. The context being alternations between simple and compound rhythms eg 3/4 and 6/8. Common in some Latin dance rhythms, especially flamenco.

Best explained with an example all us old gits are familiar with. Just try tapping your foot to the chorus of America from West Side Story.

Always a fun one to play.

Edited May 2May 2 by sethoflagos

9 hours ago, studiot said:

Yes the one and a half centenary.

My Oxford English has

sesquicentennial

Also

sesquialtera (music)

sesquioxide (chemistry)

sesquiplane (aero engineering)

sesquiterpene (chemistry)

sesquitertia (maths)

sesquipedal (organ music)

Being a musical dunce I have no idea what a hemiola is.

Hemiolas are quite common in Baroque music. Händel uses one in bars 9 and 10 of the introduction to this well-known chorus from the Messiah:

It's in 3/4, i.e. 3 crochets in a bar, so 1 2 3, 1 2 3. However in bars 9 and 10, as you approach the cadence at the end of the opening theme, the rhythmic "feel" of the music changes, like this :...... 1 2 3, 1 2, 1 2, 1 2, 1 2 3. The time signature remains 3/4 but the pattern of notes superimposes a rhythm that feels like 2-in-a-bar over the underlying 3-in-a-bar. In my score this is marked by 3 square brackets along the top and bottom of the stave, grouping the 3 pairs of the hemiola, spread across 2 bars of the music. If you listen out for it, this happens at the 10 sec mark.

It's a hemiola if it's added in - if it's a regular rhythmic change, as in "America," where there is a regular periodic shift from 6/8 to 3/4, then it's called complex rhythm. Hemiola applies to a momentary occurrence of three duple values in place of two triple ones -- sesquialtera is different, meaning a vertical two against three, like one measure with three quarters on top, two dotted quarters on bottom. Or a triplet played against two quarters. This bugs my wife, when people call sesquialteras or complex rhythms "hemiolas." My music teachers said you could call sesquialteras a "vertical hemiola" so that can give y'all an out.

A classic misnomer is calling the complex 5/4 rhythm of Mission Impossible a hemiola. It's sorta like hemiola, but again, it's baked into the whole piece so it's complex rhythm.

I find it easier in some places to grasp as 10/8 btw. 123, 123, 12, 12...

The Morse Code for MI matches the rhythm, btw. Dash dash dot dot.

https://youtu.be/XAYhNHhxN0A?si=oEQVvIbH1V9w65YO

3 hours ago, TheVat said:

My music teachers said you could call sesquialteras a "vertical hemiola" so that can give y'all an out.

Easier just to raise our eyebrows, shrug, and silently mouth the word 'Yanks' to each other (JK).

One textbook states that, although the word "hemiola" is commonly used for both simultaneous and successive durational values, describing a simultaneous combination of three against two is less accurate than for successive values and the "preferred term for a vertical two against three … is sesquialtera."[6] The New Harvard Dictionary of Music states that in some contexts, a sesquialtera is equivalent to a hemiola.[7] Grove's Dictionary, on the other hand, has maintained from the first edition of 1880 down to the most recent edition of 2001 that the Greek and Latin terms are equivalent and interchangeable, both in the realms of pitch and rhythm,[8][3]...

Perhaps rather than being relatively unschooled in the musical arts, both @exchemist and I have the Grove's on our bookshelves, rather than New Harvard. Different traditions; different words; crotchets in place of quarter notes.

4 hours ago, TheVat said:

sesquialtera is different, meaning a vertical two against three, like one measure with three quarters on top, two dotted quarters on bottom. Or a triplet played against two quarters.

Everyone I ever shared a concert platform with from age 10 onward would call this a 3:2 cross-rhythm.

A small advantage of this terminology may be some confidence that everyone understood what it meant. 😘

5 hours ago, TheVat said:

A classic misnomer is calling the complex 5/4 rhythm of Mission Impossible a hemiola. It's sorta like hemiola, but again, it's baked into the whole piece so it's complex rhythm.

... or just the simplest of the odd meters?

16 minutes ago, sethoflagos said:

Easier just to raise our eyebrows, shrug, and silently mouth the word 'Yanks' to each other (JK).

Perhaps rather than being relatively unschooled in the musical arts, both @exchemist and I have the Grove's on our bookshelves, rather than New Harvard. Different traditions; different words; crotchets in place of quarter notes.

Everyone I ever shared a concert platform with from age 10 onward would call this a 3:2 cross-rhythm.

A small advantage of this terminology may be some confidence that everyone understood what it meant. 😘

... or just the simplest of the odd meters?

I have Percy Scholes, who only provides a few lines on hemiola and if you look up sesquialtera he refers you to hemiola. There seems to be some difference of opinion in the literature about whether the terms are synonymous or not. It may be a transatlantic distinction, as with some other musical terminology, the names of grape varieties, the spelling of the chemical element Al (Americans use the original name), the numbering of seats in a rowing VIII and so much else.

3 hours ago, sethoflagos said:

Everyone I ever shared a concert platform with from age 10 onward would call this a 3:2 cross-rhythm.

Yeah, I've heard it called that, too.

It seems reasonable to see it all as semantics and not worth disbanding NATO or burning Grove's in the fire pit. I wouldn't mind just calling it all, sequential or stacked, hemiola. Hemiola is much easier to type than sesq- whatever it is. Anyway, I guess we can agree that hemiolas are incidents, while complex rhythms are regular patterns inherent to meter?

I sure don't see you or Ex as unschooled on the musical arts. And I'm a musical slacker next to the spouse and her half century as a professional musician with a double Master's and yard-long performance and teaching CV.

35 minutes ago, TheVat said:

Anyway, I guess we can agree that hemiolas are incidents, while complex rhythms are regular patterns inherent to meter?

I shall endeavour to remember that when @TheVat speaks, this is what is meant.

But what of the looser speech of others?

The song employs a mixed meter:

The alternating bars of ⁶₈ (six eighth-notes in two groups of three) with ³₄ (three quarter-notes) (similar to a guajira) is a distinctive characteristic of the song. This rhythm has been called both a hemiola and a habanera but is not really either. The two bar types alternate and are not superposed, as in a hemiola. The alternation is comparable with the "Habanera" from "Carmen", but "America" lacks the distinctive characteristic underlying rhythm of the habanera form.

I'm more than comfortable with 'mixed meter'. But some clearly do consider America to be a hemiola rhythm, and whoever last authored the Wikipedia entry doesn't even seem to know what classic sequential hemiola is anyway.

In the words of our esteemed leader ''Forget it, Jake - it's Crackpot Town". It's not science, so it's just not worth stressing out over the strictest of definitions. Few will take any notice.

My own understanding is probably based on an exercise for the hemiola in Arban's Complete Method for Cornet, which consisted solely of alternating duplets and triplets repeated indefinitely. Or at least until the pattern was imprinted forever in my marrow.

On 4/30/2026 at 6:52 PM, studiot said:

Well I have heard of 'sesqui...' in mathematics

Well I never hear of sesquation either. It is first through 3rd grade arithmetic. When you search for it on Google you have to search “sesquation in arithmetic.”

I don’t know how they teach some of these models to early elementary kids. Sometimes it seems like they should stick to how we learned when I was in school. Sometimes these learning methods are harder to understand than the actual math problem. But this one checks out and I believe it could be really powerful in a series or testing keys in cryptography.

I listed the example my friend gave me above. The formatting with the space didn’t stay making it hard to read. But Sesquation and quotation are a real thing.

We spend are our lives studying math and an elementary kids can solve patterns we can’t hope to. I believe kids can see infinite patterns they just lack the ability to describe them. The answers to the universe are there they just haven’t developed enough yet to communicate them.

In a future post I will show the example my friend gave me.

5 hours ago, exchemist said:

I have Percy Scholes, who only provides a few lines on hemiola and if you look up sesquialtera he refers you to hemiola.

I've just checked my ancient copy of the Stainer & Barrett Dictionary of Musical Terms. For sesquialtera it has:

1) Numbers in the proportion 3:2

2) An organ stop consisting of several ranks of pipes, sounding high harmonics for the purpose of strengthening the ground tone.

My money would be on this being accomplished via sum-and-difference tones, specifically the latter. Now that IS a subject worth arguing about.

2 hours ago, Trurl said:

Although I'm unfamiliar with the term "sesquation", I am aware of the concept. This is reminiscent of the notion of fractional calculus and the fractional Fourier transform. And the gamma function:

[math]\displaystyle \Gamma(z) = \int_{0}^{\infty}\!t^{z-1}\,e^{-t}\,dt[/math]

which can be regarded as a fractional form of the factorial function:

[math]\displaystyle \Gamma(n) = (n-1)![/math]

Interestingly, although there are an infinite number of functions [math]\displaystyle f(x)[/math] that satisfy:

[math]\displaystyle f(x+1) = x\,f(x)\ \ \text{for all}\ \ x > 0,\ \ f(1) = 1[/math]

the gamma function is the only such function that is logarithmically convex (see Bohr-Mollerup theorem).

7 hours ago, Trurl said:

When you search for it on Google you have to search “sesquation in arithmetic.”

Thank you yes I've seen the search results and it strikes me that this is an example of yet another person (not you) too lazy to create proper terminology for their new idea, so they pinch an older term and try to redefine it.

Nevertheless you have provided the forum with a fruitful source of discussion so +1.

You concentrate on 'sesquation', but what about quotation ?

I would say this provides a far more controvertial redefinition of a word, wouldn't you ?

Aluminium forms a sesquioxide
Al₂O₃.
Each aluminium has one and a half oxygens combined with it.
Similarly cobalt forms a sesquisulphide
Co₂S₃
And so on.

There's a joke use of the prefix in the word "sesquipedalianism"
The practice of using long and obscure words in speech or writing. Derived from Latin for "a foot and a half long,"
But when the Romans used the foot as a unit of length, they called it a pes, so I think it's "fake Latin".

21 minutes ago, John Cuthber said:

Aluminium forms a sesquioxide
Al₂O₃.
Each aluminium has one and a half oxygens combined with it.
Similarly cobalt forms a sesquisulphide
Co₂S₃
And so on.

There are also sesquiterpenes with molecular formula C₁₅H₂₄. A monoterpene has molecular formula C₁₀H₁₆ and consists of two isoprene units (C₅H₈). Thus, a sesquiterpene consists of three isoprene units, hence the "sesqui" prefix.

I found this YouTube video about "Hyperoperations" which may be of interest. I especially found the "Commutative hyperoperations" at around 16:32 to be interesting.

13 hours ago, sethoflagos said:

I've just checked my ancient copy of the Stainer & Barrett Dictionary of Musical Terms. For sesquialtera it has:

1) Numbers in the proportion 3:2

2) An organ stop consisting of several ranks of pipes, sounding high harmonics for the purpose of strengthening the ground tone.

My money would be on this being accomplished via sum-and-difference tones, specifically the latter. Now that IS a subject worth arguing about.

I think you get a third and a fifth harmonic. 12th and 17th pipe ranks over the foundation tone...OSLT. Reinforces the natural partials of the fundamental. I.e. reinforcement of the natural overtone series, for a brightened timbre. Seems like you get an additive synthesis tonal thing, not sum-and-difference tones (I mean, some S+D could trigger in your auditory perception but I don't think that's intended with a sesqui stop?) I am most def not trying to pull any rank here - I will have to check with Ms Vat later who is, fittingly, playing a church KB as I write this.

Should we move this delightful subject worth arguing about to its own thread?

2 hours ago, TheVat said:

Should we move this delightful subject worth arguing about to its own thread?

Perhaps one of the mods could perform the surgical extraction?

4 hours ago, KJW said:

I found this YouTube video about "Hyperoperations" which may be of interest. I especially found the "Commutative hyperoperations" at around 16:32 to be interesting.

Just today, I was fumbling for the word 'tetration' as a binary operation. Thank you.