A Tutorial In Working p-Adic Math Problems

 

p-adic triangles:

What are the p-adics? They're actually a specialized class of numbers first uncovered by Kurt Densel in the late 1800s. A key aspect is their absolute value, which depends on the prime number on which any p-adic is based. The p then means the particular prime. Given primes of 2, 3, 5, 7 - for example, one can find 2-adic, 3-adic, 5-adic and 7-adic absolute values which are always computed by taking the reciprocal of the highest multiple of p which divides into any given natural number, N.

If N has no multiples of p, then the absolute value is simply 1. If we are looking at a p-adic absolute value of zero, the result is always zero. (E.g. [0] _p = 0)

Let's look at some examples before examining more elaborate applications. Consider the 3-adic versions of: 7, 5 and 1/3. What will the 3-adic absolute values be? Compute each in turn:

[7] ₃  = 1 (since there are NO multiples of 3 to form the number N = 7)

 [5] ₃  = 1 (for the same reason, thus: [7] ₃ = [5] ₃ )

 [1/3 ] ₃ =  [1] ₃ /  [3] ₃ = 1/ (1/3) = 3

Since the reciprocal of 3 is 1/3 which we then divide into the numerator 1.


How about obtaining the p-adic absolute values for each of these?

[4] ₂

[1/ 6 ] ₂

[1/ 8 ] ₃ 

[24 / 25 ] ₂

The first is pretty easy, since: [4] ₂  =   [2 x 2] ₂ = 1 /4

The next isn't  terribly difficult either:

[1/ 6 ] ₂  =  [1 ] ₂   /   [3 x 2  ] ₂ = 1/(1/2) = 2

and: [1/ 8 ] ₃   = [1 ] ₃   / [8 ] ₃  = 1/  1 = 1

Note in this case, since the denominator (8) has no 3-factors, it must be that  [8 ] ₃ = 1

Lastly:

[24 / 25 ] ₂  = [3 x 2 x 2 x 2] ₂   /  [25] ₂ = (1/8)/ 1 = 1/8

(Again, 25 has no multiples of 2 which can compose it, so [25 ] ₂ = 1)

Even more intriguing are the spatial relations and differences, divergences between normal space and p-adic space. Consider the triangle (scalene) shown in Fig. 1 and the linear dimensions (absolute values) of its respective sides. We find: A = 4 ([4 - 0]); B = 6 ([10 - 4]); and C = 10 ([0 - 10]). Now compute the sides using 2-adic absolute values (I will assume the reader can perform the end computations based on the previous examples):


for A: [0 - 4] ₂ = 1 / 4

for B: [4 - 10] ₂ = [1/4 - 1/2] = 1/4

for C: [0 - 10] ₂ = [10] ₂ = 1/2

Amazingly, in the p-adic context we find the counter-intuitive result that side A equals side B. In other words, in this context, the triangle is found to be isosceles! A general rubric is that for any such computations of the p-adic absolute values contingent on a given triangle's sides - there will always be found an isosceles triangle - irrespective of how the triangle appears in normal space.

Even more bizarre results await when we examine apparently infinite series in the p-adic context. Thus, a series that first appears to go on to an infinitely large extent may be found much more different when p-adics enter. Consider the series given by a sum:

S = 1 + 5 + (5) ²  + (5) ³  + (5) ⁴  + (5) ⁵  + (5)  ⁶  + .........

To treat S p-adically, multiply both sides by 5, then place the result under the original S and subtract:

->

S = 1 + 5 +  (5) ²  + (5) ³  + (5) ⁴  + (5) ⁵  + (5)  ⁶  + .+ .........

5 S = 5   +  (5) ²    + (5) ³   + (5) ⁴  + (5) ⁵ + (5)  ⁶  + .+ .........

______________________________________________

S - 5S = 1 (all other terms above and below cancel out!)

So: -4S = 1 and S = -1/4

In other words, the sum S is less than 1 in the p-adic form - totally counter-intuitive! We see that evidently the notion or concept of "closeness" emerges quite differently - certainly if we can turn an "infinite" (apparently) sum into one yielding a result less than one!

Suggested Problems:

1- For the triangle in Fig. 2, use 7-adic absolute values applied to the sides of the triangle, thereby compute: A, B and C and show it is isosceles.

2- Find the value of the sum S for: S = 1 + 7 +  (7) ²    + (7) ³   + (7) ⁴  + (7) ⁵  + .......

3- Using (2) as written, "invent" a new irrational number based on the p-adic form.

'Adjusted GPA' - Probably The Most Logical Way To Beat Grade Inflation

 

         Loyola Theology final 1964. No student evals, no grade inflation 

Can Harvard students deal with an 'adjusted GPA' as a solution to the issue of grade inflation, which has run rampant at that illustrious university? This is a pertinent question given some months earlier they went ballistic when Harvard finally opted to cap A's. ( Wall Street Journal article ('A Harvard Cap On A's Has Students Smarting', p. A3, April 4). Noting:

For years, Harvard has been handing out A's in abundance. Now, a proposed cap would pump the brakes-and students are up in arms. Harvard's faculty is set to vote next week on a proposal to cap the number of A's per course, which now make up more than half of undergraduate grades after years of inflation.

But now, a novel method of controlling the number of A's could be the answer, proposed by WSJ guest columnist Neetu Arnold ('Whip Grade Inflation Now', 6/17, p. A13). We learn:

"There's a way to address the grading system directly while preserving faculty autonomy by adding an inflation adjusted GPA on student transcripts.

A student's transcript would show two overall grade point averages. Next to the traditional GPA will appear a second number which adjusts course grades based on the median grade in each class. A student who earns an A in a course where the median grade is also an A would receive less of a boost than one who earns an A in a tougher course.

If a student filled his schedule with easy A classes, his transcript might show a 'traditional GPA' of 4.0 and an adjusted GPA of 2.7.  This large difference communicates that the student's traditional GPA overstates his academic performance."

Of course, there is no way to know whether such a proposal wouldn't also generate howls and cries of "too much paper work, too much complexity" - not to mention; "Hey! I earned that 4.0 I don't want it mucked up by grading on a curve!"

And he or she might be correct if grades were given straight, and not deformed by the fear of student evaluations. But since the latter still dominate at most universities, to judge which professors merit promotion or tenure, we cannot just assume that the grades given aren't based on fear of a poor evaluation.

The fear factor (or its absence) can either determine giving a solid, no-holds barred mid-term or final exam, OR an easy peasy cakewalk exam.  In order to generate mor A's. If an exam is proper for a given level -  including a 3rd year Harvard course in astrophysics or economics- it should possess considerable discriminatory power.  So, even in a class of 30 Harvard over-achievers it ought to be able to separate out the 10 percent or so who demonstrate peak excellence and really merit As. This as opposed to having such an absurdly easy test that 40 percent or more get A's off it. 

Wharton School (Univ. of Pennsylvania) Professor Adam Grant didn’t like this and wrote in one NY Times review piece some years ago:

"The more important argument against grade curves is that they create an atmosphere that's toxic by pitting students against one another.  At best, it creates a hyper-competitive culture, and at worst it sends students the message that the world is a zero sum game."


Pitting students against each other?  Didn't that have to occur to show they were Harvard material in the first place? I hate to break this to our Wharton School prof but the world does operate according to a zero sum game. More to the point as Arnold writes in her WSJ piece:

"The risk of seeing such stark GPA discrepancies on transcripts would make it untenable for students to demand higher grades. Students would know that pressuring professors for grades they don't deserve could backfire.  High achieving students would also have an incentive to avoid overly lenient classes - or even encourage professors to grade more rigorously."

Adding:

"An adjusted GPA system would redirect student and institutional pressure on professors away from lenient grading toward rigor. Lenient graders rather than rigorous ones would have to adapt."

What's the big deal? It's because the 'lenient' professors are the ones terrified of bad student evaluations, which then tends to warp the grading standards across the board. To quote one physics professor in an article (Teacher Harassment and Loss of Respect') from the Aug.2020 Physics Today:

One protocol I've always disliked was the written student evaluations of professors.  A strong correlation holds between students earning low marks in physics and the ones submitting unfavorable remarks.

On the flip side, students who excel in rigorous, challenging courses will finally get their due, in terms of receiving a boost in their adjusted GPA. As the WSJ writer notes:

"Today their outstanding performance is lost in a sea of A's."

For the sake of educational authenticity as well as making the grading process transparent to employers and graduate admissions officers, we need to reduce that 'sea of A's' to a stream.

See Also:

Opinion | Harvard capping As would combat grade inflation epidemic in higher ed - The Washington Post

Excerpt:

Like monetary inflation, runaway A’s in higher education are a collective-action problem. About two-thirds of grades at Harvard College last school year were A’s. That doesn’t count A-minuses, which were another 18 percent, meaning fewer than one in six grades were a B-plus or lower.

You might have guessed grading at Ivy League schools was lenient, though not this lenient.

There’s a thoughtful solution on the table. Unfortunately, amid a student revolt last week, Harvard’s faculty postponed a vote to impose a cap on A’s. Forging ahead with the plan anyway would send a promising signal about merit and competition in American higher education.

Grade inflation — like the inflation of a currency — is a collective action problem. Professors increase the share of A’s they hand out because they know other professors are doing so and breaking from the herd would have costs. Just 35 percent of grades at Harvard were A’s in the 2012-2013 academic year, but the number climbed at a rapid clip and then surged during the covid pandemic.

The result is a collapse in the informational value of grades, especially at the high end. “As GPAs accumulate against the wall of 4.0,” a Harvard faculty committee report noted earlier this year, “the small numerical differences that remain are less reflective of genuine variation in academic performance than random noise in the grading process.”

The proposal under consideration would cap the share of A’s an instructor can give to 20 percent of the class plus four students. That means that in a large introductory course, the share of students who could get A’s — 24 out of 100, for example — would be lower than in smaller courses, which tend to be more advanced. Up to eight A’s would be available in a class of 20.

This effort matters because Harvard has the stature to prompt similar changes across the rest of higher education, where grade inflation has also been rampant. Princeton and Wellesley both tried to respond to grade inflation with caps but abandoned their efforts in 2014 and 2019, respectively.

A major objection from students at Harvard is that going back to grading on a curve will discourage them from participating in extracurricular activities. But the core purpose of campus life is learning, not socializing or networking, and academics have been excessively devalued at Harvard in recent decades. This would help restore the balance.

And:

Thanks To Grade Inflation University 'Cum Laude' Honors Are Now Meaningless

The 'Local Distance Network' Path Toward A More Accurate Hubble Constant

 

                   Artist's conception of the 'ladder' technique to obtain H₀

For scientific journal papers, few titles can compete in power and heft with:

The Local Distance Network: A community consensus report on the measurement of the Hubble constant at ∼1% precision

Which was published in the April 26 issue of Astronomy & Astrophysics (Vol. 708). In the paper 38 authors contributed to a detailed investigation exploring the so-called ‘Hubble tension’ and the degree to which two competing domains for the rate of expansion of the universe can be reconciled.  Different techniques for measuring what is called the Hubble Constant — the rate at which the universe is expanding — keep coming up with different estimates. Lead author of the University of Chicago paper, Wendy Freedman, said the Hubble Tension may be no big deal, that the scientific community may merely need to improve the way it estimates the distances between galaxies and thus the speed they recede from one another. If that happens, the Hubble Tension could be resolved without the requirement of revolutionary theories.

The Hubble tension refers to the two main ways cosmologists measure the Hubble constant (H₀), which represents the universe’s expansion rate.  (The ‘tension’ arises because the two methods don’t agree)

The method that uses the cosmic microwave background (CMB), or the leftover radiation from the Big Bang, has the constant at 41 or 42 miles (67 or 68 kilometers) per second per megaparsec (a unit of distance about 3.3 million light-years). The other approach that uses local observations of galaxies and supernova puts it at 45 miles (73 kilometers per second per megaparsec).

From their paper, we see the end result:

"The local H₀ is robustly determined, with first-rank indicators internally consistent within their uncertainties. The baseline result is H₀ = 73.50 ± 0.81 km s⁻¹ Mpc⁻¹, 7.1σ from the early Universe plus ΛCDM result 67.24 ± 0.35 km s⁻¹ Mpc⁻¹ and 5.0σ from BBN+BAO within a flat ΛCDM DESI DR2 (68.51 ± 0.58 km s⁻¹ Mpc⁻¹)."

The key component for achieving the result is the 'local distance network' - with the graphic (and codes) below from the paper:

Basically, dozens of distance indicators were assembled from dozens of teams of researchers to arrive at a coordinated product and conclusion. In a way the process could be analogized to building a 'ladder' (see top graphic from WSJ) to achieve the final result. In words of the authors:

The various subsets of the astronomical community working on aspects related to the Hubble tension have interacted at different junctures and have a general understanding of each other’s methodologies. However, accurately and reliably combining results while considering all interdependencies requires a hands-on, collaborative approach and a careful and thorough treatment rooted in transparency, engagement, and scientific discourse.

This was the underlying motivation and raison d’être for the workshop¹ “What’s under the H0od?” held at the International Space Sciences Institute (ISSI) in Bern, Switzerland, in March 2025. The goal of this workshop was to arrive at a consensus set of "baseline" and "variant" datasets that should be included, to define statistically rigorous analysis procedures that account for datasetnt covariance, and to begin developing the open access tools required to measure H₀ within a networked formalism. The Local Distance Network (Fig. 1) extends the distance ladder concept “horizontally” by linking multiple overlapping calibration paths.

In other words, this 'laddering' of different methods to reach an accurate value for  H₀ was one of the largest, most demanding cooperative astronomical ventures ever undertaken to finally try to nail down the uncertainties in the Hubble constant.  The remaining bugbear?  The end result is about 10 percent faster than what the standard model of cosmology says it should be.

This means there is probably something missing from the standard model, or a force we don't fully understand.  This according to study co-author Stefano Casertano, quoted in the Wall Street Journal. Two likely culprits here are dark matter and dark energy.

Readers can access the full paper here:

https://doi.org/10.1051/0004-6361/202557993

See Also:

• Dark Energy - Evidence for a New Law of Physics?
• And:
• All Experts Redux: The Cosmological Basis Of Dark Matter, Dark Energy

      And:

       Gravitational Lensing Exposes Further Discrepancy in Hubble Constant - Past 5 Standard Deviations

Denver Post Letter Writer Summarizes The Best Reasons To Rally Around Graham Platner

                                Graham Platner: Dems' best hope to take back Senate?

 The lead WSJ piece from June 8 (‘Platner Supporters Unfazed by Allegations of Misconduct’) proclaimed:

"Graham Platner held a town hall Sunday evening where voters could ask him about allegations of past misconduct. No one did.

Instead, voters doubled down on their support for the Democrat. The Democrat Platner teared up when presented with a poster board that included supportive messages like ‘Everyone has a past!’"

And why not? Trump certainly has a past, including admitted pussy grabbing, and having sex with a porn star even when wife Melania was caring for a newborn, e.g.

As well as being found guilty with 34 felony convictions for a hush money case connected with the affair, i.e.

Not to mention, already having incited an insurrection and attack on the Capitol on January 6, 2021,

And yet you're going to tell me, a la another WSJ piece (June 13-14, p. C3):

"Maine's Platner faces an uphill battle in the wake of his sexting scandal."

When the orange swine fouling the White House has gotten away with not only openly bragging about grabbing female genitals, actual sexual assault on E. Jean Carroll, not to mention palling around with a pedo predator- and even sending him a lascivious birthday card,

 WTF is wrong with American brains when a tattoo and some ill-advised ribald riffing on Reddit can make it "uphill" for Platner but Donnie Bonespurs gets a free pass?

Again, none of the cited outrageous actual crimes of Trump have had any effect on stopping him - having being elected twice - and even getting over 74 m votes in his losing effort to Biden in 2020. And yet Graham Platner is dead in the water for 'sins' that are more like crass verbal missteps- compared to Trump's mammoth crimes - like blowing up hundreds of innocents in fishing boats in Caribbean waters. 

And yet Traitor Trump had the nerve to call Platner a "thug" and a "pig". Hey, traitor -convicted felon- pedo pal - pussy grabber, look in the mirror.

Enter a Sunday Denver Post letter writer who throws down the political purity breaking gauntlet and defies Dems to find their cojones and support Platner. He writes in his letter ('Dems Should Close Ranks Around Platner):

Moral high ground? Elections are not about morals.  They are about power - who gains power, how they are exercising it and accountability for that power.

Elections are not dating events. Voters do not choose a mate. A pure heart. an unimpeachable background, perfect manners and a dental plan do not matter.  Graham Platner is not running to be Maine's sweetheart. He is running for one of its Senate seats. Platner is Maine's Democratic Senate candidate.

There is no other. It is him against Republican Susan Collins who talks centrist and votes extremist. He is the Democratic candidate because he has been leading in the polls and offers the party a chance to beat Susan Collins. Democracy in action.

For the sake of us, the people, our country and the planet, Democrats must now regain power. Only with power can they combat the corruption and the chaos, legislate for the people and reverse the prevailing pattern of offensive and vulgar conduct - which columnist Doug Friednash sanctimoniously and wrongfully attributes to Platner, but which the corrupt Trump regime inflicts on we the people, every day and all day long.

Platner with a covered up Totenkopf tatoo? Cute considering Defense Secretary Pete Hegseth uncovered vile white supremacist tattoos and activities. Foul language? Who cares in light of Trump and Co's incessant sewage tweets. Imp[ure manners towards women who are not his wife. Commanding them to 'be quiet, piggy' or bragging about grabbing their nether regions without consent.

Those who want Dems to gain power must close ranks behind Dem candidates, not sabotage them with irrelevant purity tests.

The letter writer is correct on so many counts, I literally could see the finished product as a submission I might have written myself. What is so gob smacking hypocritical is how so many mainstream media Nanny-ites could be picking Platner to shreds yet turn a blind eye to all of Trump's post-midnight howls,

Open the fucking Strait, you crazy bastards, or you'll be living in hell. JUST WATCH! A whole civilization will die tonight

as well as actual crimes from sexual assault to inciting an insurrection and paying hush money to a porn star. (To keep quiet about their trysts before the 2016 election). Not to mention being a Putin asset and launching a war of choice that will have horrific consequences on the global economy for months, if not years.

"Yeah, but that's just Trump, we know what he's like and expect it!"

Well, tough shit. Because if the Maggats are going to vote for a convicted criminal like Trump the rest of us seeking to curb his power  (unsurpassed because of cowards like Susan Collins) then we must back any Dem candidate with less toxic negatives to turn the tables. And without turning into lily-livered fusspots like much of the media has. 

In other words, with so much on the line, we cannot afford to be choosy. We don't have that luxury. Yes, as a recent (June 8) TIME article put it:

 "Platner is newcomer carrying enough baggage to sink an oyster boat"

But Trump is an actually elected felon and traitor who's amassed enough REAL crimes to make Hitler blush.

Finally, let's bear in mind Capt. Bonespurs dodged the draft five times. Graham Platner (as the TIME article notes):  

"Served three tours as a Marine infantryman, engaging in close combat in Ramadi and Fallujah."

I will take an honest, up front former Marine - though his verbiage be on the rough side - before a 5-time draft dodger, 34 time convicted felon and insurrectionist traitor.  Who once referred to troops killed in action as "losers."

See Also:

2026 Elections | Democrats

Despite Media Knocks, Maine’s Graham Platner Qualifies As This Year’s Breakthrough Senate Campaign Star

by Robert Becker | June 15, 2026 - 4:57am | permalink

Imagine a charismatic, working class progressive with strong policy values, down-home personality, and awesome articulation – displaying the self-esteem and honesty to apologize and seek forgiveness.

It’s dicey to endorse national candidates without either state or Washington experience. Heading his local Planning Board, salvaging an oyster farm and serving as town harbor master, plus military tours, capture how Platner proves good, heart-felt politics start locally. Were this independent, true “outsider” (disowned at first by tin-ear Dems) to advance to the Senate, that would be a stunning signal that change is on deck.

Here’s the ideal, “citizen” prospect who commits to “working Mainers” by joining a looming progressive coalition to overcome the dismal D.C. stall. Only a serious coalition does anything lasting – and this upstart activist wants to be Maine’s coalition voice. That he looks good to flip a Senate seat grabs everyone’s attention.

» article continues...

And:

Why Do White Christian Evangelicals Worship A Pervert Prez Who Loves Spankings & Golden Showers?

And:

Donald Trump

Trump’s “White Working Class” Base May Finally Have Had Enough

by Heather Digby Parton | June 17, 2026 - 5:11am | permalink

— from Salon

Donald Trump threw himself a great big birthday party over the weekend, targeted to appeal to the famous white working-class voters who have formed the core of his base since the moment he came down that golden escalator. He transformed the White House lawn into a makeshift Colosseum and held a modern-day gladiatorial spectacle, officially called UFC Freedom 250. One participant vomited on himself and later made offensive comments about Michelle Obama, to the great amusement of the assembled crowd. But there are signs that this sort of cultural signifier may not be enough to keep the MAGA faithful on board anymore.

It’s always been a bit strange that this spoiled, Richie Rich-style heir to a real estate fortune could possibly become an avatar of the working class. He’s never known a moment of physical labor in his life and has nothing but contempt for any of the hard-fought regulations and rights that protect workers from the predations of the moneyed elite and give them a chance at the American dream. One might have thought that all his ostentatious displays of wealth, compared to the squeeze working people have faced for at least the last generation, would at least make them skeptical. But as we’ve all discussed ad nauseam over the last couple of decades, tribal grievance tends to trump material self-interest among a large faction of the American electorate.

» article continues...

Advanced Astrophysics: Solutions To Stellar Emission & Absorption Problems

1) Consider a gas of neutral hydrogen. Using the Boltzmann equation and the information in the table (see April 2 blog post), compute the temperature at which one will expect equal numbers of atoms in the ground state and the first excited state.

Solution:

The Boltzmann equation is:

N₂ / N₁  =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

And from the table, g₂ = 8 and  g₁ = 2

We require the condition that:  N₂ =  N₁     so:

1 =     [8/ 2 ] exp (- E2 – E1) / kT

But:  E2 =  - 13.6 eV and E1 = -3.4 eV, therefore:

1 = 4 exp [- 13.6eV – (-3.4 eV) ]/ kT

1 =  4 exp (-10.2 eV)/ kT

Taking natural logs:

ln (4)  =    (10.2 eV)/ kT

where: k =   8.6174 x 10 ^-5 eV/K

Solving for T: 

T =    10.2 eV/  (ln 4) (8.6174 x 10 ^-5 eV/K)

T =  10.2 eV/ (1.3862) (8.6174 x 10 ^-5 eV/K)

T= 85 388 K or T = 8.54 x 10 ⁴ K

 

       2) For the Balmer a line (called H- alpha), we know:

E3 – E2 =  - 13.6 eV ( 1/ 3 ²   -  1/ 2 ² )   = 1.88 eV

a)     From this information calculate the ratio N₂ / N₁    

b)     Obtain the specific intensity from: 

I _u  =  2h u ³ / c ²  [1/ exp (hc/lkT]

Solution:

N₂ / N₁   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

From the table in the Dec. 5th  post;  g₂ = 8 and  g₁ = 2   

k =   8.6174 x 10 ^-5 eV/K

N₂ / N₁    =     [8/ 2 ]   exp (- E2 – E1) / kT

N₂ / N₁      =  4 exp (-1.88 eV)/ (8.61 x 10 ^-5 eV/K) (10 ⁴ K)

N₂ / N₁      =  4(0.113) =  0.452

b)I _u  =  2h u ³ / c ²  [1/ exp (hc/lkT]

We need to use consistent cgs units.  Planck constant h = 6.62 x 10  ^-27 erg-s

c=  3 x 10  ¹⁰ cm/s

l  =  hc/ E =  (6.62 x 10  ^-27 erg-s) (3 x 10  ¹⁰ cm/s)/ 3.0 x 10  ^-12 erg

l  =  6.62 x 10  ^-5 cm 

k=1.38 x 10  ^-16 erg/K

[1/ exp (hc/lkT]  =

1/ [exp (6.62 x 10  ^-27 erg-s) (3 x 10  ¹⁰ cm/s)/ (6.62 x 10  ^-5 cm)  (1.38 x 10  ^-16 erg/K)( 10 ⁴ K)

= 0.113

I _u  =   2h u ³ / c ²  [0.113] erg cm ^-2/s

But  u =   E/h   =  

3.0 x 10  ^-12 erg/ 6.62 x 10  ^-27 erg-s  =  4.53 x 10  ¹⁴ /s

So:

I _u  =    0.226(6.62 x 10  ^-27 erg-s) (4.53 x 10  ¹⁴ /s) ³ / (3 x 10  ¹⁰ cm/s) ²

I _u  =     1.51  x 10  ^-4 erg cm ^-2/s

3)Calculate  the transition probability you get using the Einstein equation:

A ₂₁=   6.67 x 10 ¹⁶ [g f/g2  l²   Å]

What possible errors might cause the values to diverge? (Take g = f » 1)

  l =  6.62 x 10  ^-7 m  =  6.62 x 10  ^-5 cm =  6620 Å

With g₂ = 8  and g = f = 1

A ₂₁=   1.93 x 10 ⁸ [Å^-2]

Compare to standard form (see e.g. Wikipedia, “Einstein coefficients”)  given in multiple physics papers as:

A ₂₁=  [f g1/g2]{ 2 π  u ³  e ² }/ e _o  m_e c³  

A ₂₁ =    1.87 x 10 ⁷    or:    0.187  (in defined units of    10 ⁸  s)      

 

Error sources: Imprecise oscillator frequency f

Error in one of the statistical weights.

Uncertainty in line measurements for absorption and extrapolating this for deduction of  A _21.