Why I Love Lichens But Oppose Nuclear Power

Parmelia sulcata, one of the lichen species from which litmus powder can be extracted

Nature has unexpected sensors. Students are always astonished to learn that litmus, the simplest of acid-base indicators, is extracted from lichens. Lichens, which have no roots but a thallus, also bioconcentrate SO2 pollution and radioactive fallout from the air. The thallus is the non reproductive part of the lichen that, depending on the genus, can have a variety of appearances as shown below.

Four different groups are growing together on this iron-rich rock. The island-like boundaries are those of the prothallus. From http://www.lichens.lastdragon.org/faq/lichenthallustypes.html

There are at least three examples of how lichens have ended up concentrating radioaoctive isotopes:

(1) In studies of fallout following atomic bomb tests in the 1950s, it was found that both Alaskan Eskimos and Scandinavian Laplanders had unusually high levels of radioactive strontium and cesium. This happened even though polar regions received less fallout than tropical areas. The cause? In polar areas, bioconcentration by lichens and subsequent bioaccumulation occurred as the isotopes moved through the food chain from caribou ( who eat reindeer moss which is a lichen) , which are then hunted by Eskimos and Laplanders.

(2) Through the same mechanism, reindeer (what Laplanders call caribou) also became unfit for human consumption after the Chernobyl accident of 1986.

(3) More recently, the concentration of cesium isotopes (¹³⁴Cs and ¹³⁷Cs ) in lichen within a 30 km radius of Fukushima were more than 3 times the acceptable limit.

The Fukushima disaster is a dim reminder of the dire consequences that nuclear energy can have, even when it’s used “peacefully”. A 20-kilometre zone around the reactors had to be evacuated. Food was contaminated and six years later, 97 000 people have still not returned to the area. This includes people who still have to be barred from the region, not just those who opted to live elsewhere. Although it seemed initially that the tidal wave was the primary cause of the nuclear accident, there was in fact serious damage to piping in at least one of the reactors before the tsunami hit them. In the 2015  film Fukushima-a-nuclear-story former Prime Minister Naoto Kan reveals that Tokyo and all of Japan was saved from a much greater catastrophe by chance when a faulty gate caused water to accidentally flow and cool the top of a reactor.

Largely in response to the Fukushima disaster, Switzerland almost voted in favour of a strict timetable for a nuclear power phaseout. And a long-term plan to shift to renewables does exist in a country which presently derives 45% of its electricity from nuclear energy.

Even though nuclear energy is hailed as being carbon-free, I have long held my reservations about the technology. Here’s why.

Where nuclear reactors are found worldwide. From euronuclear.org

 (1) There are presently no long-term solutions for the storage of nuclear wastes.

(2) Using nuclear technology to generate electricity is far more expensive than other forms.

(3)  A thorium-based technology is currently not being used, even though it would create less waste. Part of the reason it wasn’t chosen as a fuel is simply because countries have always wanted the plutonium from current uranium-based technology for weapons-purposes. But thorium’s short term radiation from its fission products is harder to contain, which would further raise the costs.

(4) Because of the magnitude of a large-scale explosion, the low probability of its occurrence will be no consolation if it actually occurs. And the likelihood will increase if nuclear power becomes more ubiquitous.

(5) The record of the industry has been far from impeccable and its attitude far from sincere. Things will not get any better with ageing reactors.

Here’s a list of countries that do not use fission reactions to boil water to turn turbines —who,  as I once heard an activist say, “don’t burn the house down to make toast.”

• Australia
• Austria
• Belize
• Cambodia
• Colombia
• Costa Rica
• Greece
• Ireland
• Italy
• Latvia
• Liechtenstein
• Lithuania
• Nepal
• New Zealand
• Peru

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The Promise of a Non-addictive Opioid

As far back as the late 1800s heroin was once hailed with the promise of being an abuse-free opiate. Although dozens of other opiates have been created, no one in the past 12o years has succeeded in creating a side-effect-free and non-addictive opioid. But by using a new approach, there is a chance that a German team led  by the anesthesiology and professor Dr. Christoph Stein has succeeded. Before delving into the specifics and exploring the chemistry, let’s provide some background information.

What Are Opioids?

An opioid can be any of these compounds: (1) it can be part of the group of substances directly made by, or derived from, the opium poppy; (2) it can be one of the human-brain-produced peptides that influence a variety of  behaviours including attachment, stress, food-intake and pain response; or (3) it can be a synthetic substance that binds to the same receptors as the previous two types. The receptors known so far include μ (mu—the m coming from “morphine”), δ (delta) and κ (kappa).

Opioids are used to treat pain brought on by terminal or serious illnesses, such as cancer. Some medical practitioners also turn to opioids for chronic pain, common among those with forms of arthritis, back injuries, torn muscles, damaged nerves and fibromyalgia. A case of kidneys stones, nature’s way of making men experience something almost as painful as childbirth, sure made me appreciate a dose of morphine. Since that experience, I have not needed a strong opiate.

Others are not as lucky and need more intense compound-assisted pain management, especially if other opioids have become ineffective. Fentanyl (trade names: Actiq^®, Duragesic^®, and Sublimaze^®) is a quick-acting synthetic opioid with 50-100 times the strength of morphine. But transdermal patches of fentanyl are prescribed only when at least a week of tolerance towards opiates has already been established. That’s because with all its delivery-forms, the use of fentanyl can be very risky. Opioid receptors are all over the body, brain included, and fentanly will not only cause dopamine levels to spike, but it could also bind to brain receptors that control the breathing rate. Respiratory depression can be fatal, and the online availability of illicit fentanyl sold through decoy packages has led to an epidemic rise in fentanyl-related deaths in Canada, Estonia and the United States.

The first pouch on the right hides a few grains of fentanyl (enough to kill). The package was mailed from China pretending to be a set of urine-testing strips. From the Globe and Mail

Other undesirable reactions from binding to μ receptors include reduced gastrointestinal nausea, vomiting, and the accompanied euphoria from the ensuing dopamine-spike, which ties into addiction.

How A Safer Alternative Was Found

The researchers started by focusing on the role of MOR (μ opiate receptors) in injured tissues as opposed to the brain. They did this for two reasons (1) Painful conditions such as inflammation or trauma are often associated with a lower pH(acidification) in localized tissue. (2) Only the protonated form of opiates like fentanyl can bond to MOR receptors. To be protonated simply means that an H+ ion has been attached to a basic group, in this case the nitrogen of a heterocyclic ring called piperidine . Then through hydrogen bonding, that proton attracts the oxygen of a carboxylate group of an aspartic acid molecule on the receptor.

The protonated fentanyl shown bonded to an aspartic acid residue (Asp 147) of a μ opiate receptor. Modified from author’s paper in Science.

Fentanyl and other opiates have substantial fractions of their molecules protonated at pHs of about 5 to 7, where the source of pain lies.  But the problem is that opiates are also highly protonated at pH 7.4 , the typical pH of the brain and small intestines where you don’t want the opiates to be activated.

We will represent the protonated form of fentanyl as HNFen⁺ and the uncharged basic counterpart as NFen:

HNFen⁺ = H⁺ + NFen

What the researchers aimed to do was to attach a fluorine group to key positions of the fentanyl molecule to increase its acidity without altering the stereochemistry or functional groups of the rest of the molecule. After some computer modelling they suggested the synthesis of NFEPP:

The protonated form of NFEPP, which has been shown to be a non addictive opioid still capable of acting as an analgesic. This form becomes less important above a pH equal to its pKa of 6.8

The protonated form of fentanyl found in the commercial citrate. It’s also the predominant form below pH = pKa = 8.4, so in other words in any human tissue containing opioid receptors.

Even at the beta position (on the second adjacent carbon attached to nitrogen), fluorine’s electron-withdrawing presence decreases a nitrogen’s ability  to reattach itself to H⁺. This happens because nitrogen uses its electron pair to act as a base and bond to H⁺. If the reverse reaction is less likely to proceed, the acidic forward reaction is favoured. That is a good thing for the drug designers because they did not want the protonated form to dominate in certain tissues where it could lead to addiction and respiratory arrest.

To understand why a more acidic molecule will be adequately protonated in the proton-rich environment —where the pain is—and why there will be far less protonated than fentanyl at pH=7.4, let’s explore some analytical chemistry.

For weak acid systems like

HNFen⁺ = H⁺ + NFen , we cannot make the assumption that the H⁺ concentration will equal that of NFen. To estimate the relative concentration of the protonated fentanyl  we have to rely on the mass balance approach where we compare HNFen⁺ ‘s concentration, [HNFen⁺ ], to that of the total of  non H+ species.

We let X = [HNFen⁺ ]+ [NFen] = sum of concentration of non-H+ species.

so [NFen] = X – [HNFen⁺ ]

since its acid dissociation constant, Ka = [H⁺][NFen]/[HNFen⁺ ], by susbtituting for [NFen] we obtain:

Ka = [H⁺](X – [HNFen⁺] )/[HNFen⁺].  By then solving for [HNFen⁺ ], we obtain:

[HNFen⁺] =  [H⁺]X / ( Ka + [H⁺] )

Finally by rearranging the expression, we get the fraction of protonated species as:

[HNFen⁺]/X = [H⁺]/ (Ka + [H⁺] )

The expression suggests that the fraction of the protonated species is pH-dependent and is inversely proportional to the strength of the acid (measured by its Ka). If [H⁺] = Ka or if pKa = pH , we will get a mole fraction of exactly of 0.5. This implies that since NFEPP’s pKa is lower (6.8) than fentanyl’s, then the concentration of the protonated form will be the minority-species at a pH that’s lower than the crossover point for fentanyl. The latter has a lower Ka and hence higher pKa of 8.4. To better reveal the relationship, I plotted the fraction [HNFen⁺]/X versus the pH for both fentanyl and for NFEPP, which due to its fluorine has a higher Ka.

Notice how the fraction of the protonated form of NFEPP drops dramatically to about 10% at pH= 7.4 whereas that of fentanyl is still above 90%. Meanwhile NFEPP still acts as an analgesic because at pHs of 6.8 to 5.2, the fraction of the protonated form that’s needed to bond to μ receptors in inflammation areas is in the 52 to 95% range, respectively.

But does all this theory work in a biological environment? In the authors’ experiments, NFEPP produced analgesia in rats who had different types of inflammatory pain. And equally important the pain relief was not accompanied by typical opiate side effects such as sedation, decreased breathing, constipation or the desire to seek more pain-killer. In 4 to 5 years, the time it will take to refine the synthesis and test the product in humans, it will be interesting to see if the usual unforeseen consequences will be minor. Of course NFEPP ‘s inability to produce euphoria will not displace dangerous opiates from the black market. But it will help alleviate the problem by putting nonaddictive drugs into circulation while hopefully making the legitimate production and prescription of fentanyl and other opiates obsolete.

Sources:

A nontoxic pain killer designed by modeling of pathological receptor conformations http://science.sciencemag.org/content/355/6328/966

Fluorine in Medicinal Chemistry ChemBioChem 2004, 5, 637 ± 643 https://ccehub.org/resources/611/download/637_ftp.pdf

Click to access BBD-Opioids-Full.pdf

A brief history of opiates, opioid peptides, and opioid receptors Michael J. Brownstein Laboratory of Cell Biology, National Institute of Mental Health, Bethesda, MD 20892  http://www.pnas.org/content/90/12/5391.short

http://www.sciencedirect.com/science/article/pii/S0304395904004476

https://www.drugabuse.gov/publications/drugfacts/fentanyl

http://www.theglobeandmail.com/news/investigations/a-killer-high-how-canada-got-addicted-tofentanyl/article29570025/

http://www.theglobeandmail.com/news/national/estonias-cautionary-fentanyl-tale-for-canada/article30512811/

Why Perfect Numbers Only End in 6 or 8

The search for answers to simple questions like “why is the sky blue? ” and “why are there selfish acts” is a nice way to bite into a fair amount of science and philosophy, respectively. In math the same is true. Why do even-perfect numbers only end in 6 or 8? If you’re fortunate enough to have a child asking you such a question , all you need is about 1200 words to answer it—only 1/9 th of the number of words in a sitcom script. 🙂 And the discussion of perfect numbers begins with a venture into a special type of prime number.

Mersenne, a priest, was a mathematician but primarily a music theorist and a hub for scientific communication in the 1600s.

As of February 2022, of all the known primes, only 51 of them are Mersenne primes. A Mersenne prime M_n can be expressed as M_n = 2^p − 1, where p itself is prime.  The largest prime to date is 2^{82 589 933} − 1, known as M51. To put things into perspective, our universe only has 10⁸⁰ atoms, a number with only 81 digits. But M51 has 24 862 048 digits, meaning that it is bigger than the inconceivably large 10 ¹⁰⁷. Given that the number of primes less than 10 ¹⁰⁷ is approximately equal to 10 ¹⁰⁷/log(10 ¹⁰⁷), in comparison, 49 (plus the possibly few smaller Mersennes yet to be discovered), divided by that gargantuan number is an unimaginably small fraction indeed.

1. Why p must be prime if 2^p − 1 is prime

An interesting matter is why p must be prime if 2^p − 1 is prime. To convince ourselves, we will use a proof by contradiction—in other words we will make an initial assumption that states the opposite condition, logically follow it to a contradiction, and in the process reveal the assumption to be invalid.

We know from a geometric series that :

xⁿ − 1 = (x − 1)(x^{n − 1} + x^{n − 2} + · · · + a + 1)

For example x³ – 1 = (x-1) ( x² + x  + 1)
Let’s assume that the exponent was composite, such that the Mersenne prime would be expressed as 2^an – 1. In other words we assume its exponent to be composite.  Let x = 2ª. from the above geometric series; then
2^an − 1 = (2ª − 1)(2ª^{(n − 1)}+ 2ª^{(n − 2)} + · · · + 2ª + 1)

But now we have expressed a prime number as two factors, neither of which is equal to one, which is impossible. The original assumption about p being composite was therefore false, so p must be prime.

2.  Getting a perfect number from a Mersenne Prime

Next we will multiply the Mersenne prime by a power of 2 whose exponent is one less than the prime used to generate the Mersenne prime. In other words:

(2^p-1 )(2^p − 1).

Interestingly this resulting number will always be a perfect number, a number that is the sum of all of its factors except itself. For example 28 is perfect because ( 2^3-1 )(2³ − 1) = 1 + 2+ 4+7 + 14 = 28.

To reveal the “perfection” of any number of the form (2^p − 1) ( 2^p-1 ) where p is prime, we remind you of the function σ(x).  σ(x) is equal to the sum of all the factors of x including x. Adding a perfect number, N= ( 2^p-1 )(2^p − 1), to a sum that already equals N, implies that in order for a number to be perfect,

σ(N) = 2N =  2 ( 2^p-1 )(2^p − 1)= ( 2^p )(2^p − 1) … (equation 1)

We start with the theorem σ(ab) = σ(a)σ(b), where a and b’s greatest common divisor =1. The reason this theorem is valid is that multiplying the sum of factors of two individual factors a and b is equivalent to getting all the non-redundant combinations of the product’s factors. For example, consider ab = 12 where σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. That is equivalent to doing the following : σ(4)σ(3) = ( 1 + 2 + 4)(1 + 3) = 1(1) + 1(2) + 1(3) +2(3) + 4(1) +4(3) = 28.

Applying the theorem to our expression:

 σ[( 2^p-1 )(2^p − 1)] = σ( 2^p-1 ) ∗ σ(2^p − 1).                    … equation (2)

Beginning with σ( 2^p-1 ), we have here simply the sum of the factors of a power of two up to that power. σ( 2^p-1 ) = 2⁰ + 2¹ + 2² + 2³ + …..2^p-2 + 2^p-1. 

We add 1 to each side:

1+ σ( 2^p-1 ) =( 1 + 2⁰ )+ 2¹ + 2² + 2³ + …..2^p-2 + 2^p-1. 

We notice that the first terms’ sum is equal to the next power such that:

1+ σ( 2^p-1 ) = (2¹  + 2¹ )+ 2² + 2³ + …..2^p-2 + 2^p-1.  But this keeps happening:

1+ σ( 2^p-1 ) = (2²  + 2² )+ 2³ + …..2^p-2 + 2^p-1.  When we get to the second last term, we will have

1+ σ( 2^p-1 ) = (2^p-2 +2^p-2 )+ 2^p-1.

1+ σ( 2^p-1 ) = (2^p-1 )+ 2^p-1 =  2^p ,

or σ( 2^p-1 ) = 2^p – 1                                                       ……equation (2b)

We substitute this result into equation (2) and obtain

 σ[ ( 2^p-1 )(2^p − 1)] = (2^p – 1 ) * σ(2^p − 1).                          … equation(3)

But σ(2^p − 1) is the sum of the factors of a prime number which means that we merely have to add 1 to itself = 1 + 2^p − 1 = 2^p . 

Substituting this result into equation (3) we obtain:

 σ[( 2^p-1 )(2^p − 1)] = (2^p – 1 )2^p , which meets the condition we had outlined for a perfect number in equation (1)!

3. Why multiply a Mersenne by ( 2^p-1 )?

A derivation of S = n(n+1)/2

But where does the idea of multiplying a Mersenne prime (2^p − 1) by ( 2^p-1 ) come from? To get the sum, S,  of all the natural numbers from 1 to 100 in other words (1 + 2+ 3+ 4 ….100), we simply multiply the largest number by the next natural number and divide by 2, so S(100) = 100*101/2 = 5050. The formula is S(n)= n(n+1)/2 and I derived it informally and partly pictorially in the adjacent drawing.

Applying the formula to the Mersenne prime we get (2^p − 1)(2^p − 1 + 1)/2 = (2^p − 1)(2^p)/2¹ = (2^p − 1)(2^p-1)

4. But does an even perfect number have to be of the form (2^p − 1)(2^p-1) ?

The answer is yes, and here is my summary of a clearer version of the Euclid-Euler Theorem found here. Let the even perfect number = 2^k  *a, where a is an odd number.

As seen earlier, if the number is perfect then

σ(2^k *a) = σ(2^k) σ(a) , which applies since a power of 2 and an odd number will never have a common factor greater than 1

As we demonstrated to get to equation 2(b), the sum of the factors of a power of 2 is equal to the next power of 2 minus 1, so

σ(2^k  *a) = 2^k+1 -1 * σ(a).                                                   ….equation (4)

Since 2^k  *a is perfect, the sum of all of its factors will also equal twice itself, so:

σ(2^k  *a) = 2*2^k  *a = 2^k+1 *a                                                  …equation (5)

Equating equations (4) and (5) we obtain,

(2^k+1 -1)* σ(a) =  2^k+1 *a                                                         …equation (6)

or rearranging  σ(a) = 2^k+1 *a /(2^k+1 -1)

We can see from this expression  that σ(a) is the product of 2^k+1 and another expression which we’ll simplify as b.

so σ(a) = 2^k+1 *b         , where b = a/ (2^k+1 -1)                …equation (7)

Rearranging the expression for b,

a = (2^k+1 -1) * b and

σ(a) = σ( (2^k+1 -1) * b)                                                               …..equation (8)

From equation (7)

σ(a) = 2^k+1  *b , which we can rewrite as σ(a) =2^k+1  *b -b + b . If we then factor the first two terms to get:

σ(a) = b(2^k+1   -1) + b .                                                                  ….equation (9)

Equating (8) and (9):

σ( (2^k+1 -1) * b)   =  b(2^k+1   -1) + b .     The only way that the factors of  (2^k+1 -1) * b  won’t exceed the discovered sum of  b(2^k+1   -1) + b is if b =1, so our original odd number a only has two factors, one of which is 1,  revealing that a is prime. Moreover a is a Mersenne prime since it is of the form 2^k+1 – 1

A contemporary of Mersenne, Pietro Cataldi discovered that perfect numbers end in either 6 or 8.

7. Finally we return to the question in the title. Why do all even perfect numbers (no one yet found an odd-numbered one ) have to end in 6 or 8?

A perfect number = ( 2^p-1 )(2^p − 1). The first factor of that expression,  2^p-1 , will always create an even power of 2, except for 2^2-1 which will still satisfy our overall generalisation by creating 6. All even powers of 2 end either with the digit 6 or 4. The next power 2^p , which follows 2^p-1 , ends in either 2 or 8, respectively. But we subtract 1 from each of those numbers because we are multiplying ( 2^p-1 ) by (2^p − 1). That translates into either multiplying a number ending in 6 by (1=2-1), which will end in 6 or a number ending in 4 by (7= 8-1), whose result will end in 8.

Here’s a table from Wikipedia of all known perfect numbers as of February 16, 2022. To date, thirty two of the discovered perfect numbers end in 6 and nineteen end in 8.

Rank	p for Mp	Perfect number	Digits	Year	Discoverer
1	2	6	1	4th century B.C.[5]	Euclid
2	3	28	2	4th century B.C.	Euclid
3	5	496	3	4th century B.C.	Euclid
4	7	8128	4	4th century B.C.	Euclid
5	13	33,550,336	8	1456	First seen in a medieval manuscript, Munich, Bayerische Staatsbibliothek, CLM 14908, fol. 33[6]
6	17	8,589,869,056	10	1588	Cataldi[1]
7	19	137,438,691,328	12	1588	Cataldi[1]
8	31	2,305,843,008,139,952,128	19	1772	Euler
9	61	265845599156…615953842176	37	1883	Pervushin
10	89	191561942608…321548169216	54	1911	Powers
11	107	131640364585…117783728128	65	1914	Powers
12	127	144740111546…131199152128	77	1876	Lucas
13	521	235627234572…160555646976	314	1952	Robinson
14	607	141053783706…759537328128	366	1952	Robinson
15	1,279	541625262843…764984291328	770	1952	Robinson
16	2,203	108925835505…834453782528	1,327	1952	Robinson
17	2,281	994970543370…675139915776	1,373	1952	Robinson
18	3,217	335708321319…332628525056	1,937	1957	Riesel
19	4,253	182017490401…437133377536	2,561	1961	Hurwitz
20	4,423	407672717110…642912534528	2,663	1961	Hurwitz
21	9,689	114347317530…558429577216	5,834	1963	Gillies
22	9,941	598885496387…324073496576	5,985	1963	Gillies
23	11,213	395961321281…702691086336	6,751	1963	Gillies
24	19,937	931144559095…790271942656	12,003	1971	Tuckerman
25	21,701	100656497054…255141605376	13,066	1978	Noll & Nickel
26	23,209	811537765823…603941666816	13,973	1979	Noll
27	44,497	365093519915…353031827456	26,790	1979	Nelson & Slowinski
28	86,243	144145836177…957360406528	51,924	1982	Slowinski
29	110,503	136204582133…233603862528	66,530	1988	Colquitt & Welsh
30	132,049	131451295454…491774550016	79,502	1983	Slowinski
31	216,091	278327459220…416840880128	130,100	1985	Slowinski
32	756,839	151616570220…600565731328	455,663	1992	Slowinski & Gage
33	859,433	838488226750…540416167936	517,430	1994	Slowinski & Gage
34	1,257,787	849732889343…028118704128	757,263	1996	Slowinski & Gage
35	1,398,269	331882354881…017723375616	841,842	1996	Armengaud, Woltman, et al.
36	2,976,221	194276425328…724174462976	1,791,864	1997	Spence, Woltman, et al.
37	3,021,377	811686848628…573022457856	1,819,050	1998	Clarkson, Woltman, Kurowski, et al.
38	6,972,593	955176030521…475123572736	4,197,919	1999	Hajratwala, Woltman, Kurowski, et al.
39	13,466,917	427764159021…460863021056	8,107,892	2001	Cameron, Woltman, Kurowski, et al.
40	20,996,011	793508909365…578206896128	12,640,858	2003	Shafer, Woltman, Kurowski, et al.
41	24,036,583	448233026179…460572950528	14,471,465	2004	Findley, Woltman, Kurowski, et al.
42	25,964,951	746209841900…874791088128	15,632,458	2005	Nowak, Woltman, Kurowski, et al.
43	30,402,457	497437765459…536164704256	18,304,103	2005	Cooper, Boone, Woltman, Kurowski, et al.
44	32,582,657	775946855336…476577120256	19,616,714	2006	Cooper, Boone, Woltman, Kurowski, et al.
45	37,156,667	204534225534…975074480128	22,370,543	2008	Elvenich, Woltman, Kurowski, et al.
46	42,643,801	144285057960…837377253376	25,674,127	2009	Strindmo, Woltman, Kurowski, et al.
47	43,112,609	500767156849…221145378816	25,956,377	2008	Smith, Woltman, Kurowski, et al.
48	57,885,161	169296395301…626270130176	34,850,340	2013	Cooper, Woltman, Kurowski, et al.
49	74,207,281	451129962706…557930315776	44,677,235	2016	Cooper, Woltman, Kurowski, Blosser, et al.
50	77,232,915	109200…301056	46,498,850	2018	Cooper, Woltman, Kurowski, Blosser, et al.
51	82,589,933*	110847…207936	49,724,095	2018	Cooper, Woltman, Kurowski, Blosser, et al.

The cover and an extract from Cataldi’s treatise on perfect numbers. The entire original is found at http://mathematica.sns.it/media/volumi/69/Trattato%20de’%20numeri%20perfetti_bw_.pdf