Tropical Storm Erin (2019)

Storm Active: August 26-29

On August 21, an trough of low pressure near the eastern Bahamas began to produce a scattered area of shower and thunderstorm activity. It moved slowly west-northwestward, but developed only slowly due to wind shear. Just as the storm became more concentrated on August 23, the newly formed low pressure center moved over southern Florida. The land interaction temporarily hampered further progress toward tropical depression status and most rainfall remained offshore to the east. Within another day, the system felt the tug of an approaching cold front and turned northeast, emerging over water once again. The low pressure center remained elongated, though, and westerly shear kept the western half of the circulation dry. Nevertheless, Tropical Depression Six formed during the afternoon of August 26, well offshore of North Carolina. At that time, Six had slowed to a near standstill due to the influence of nearby mid-level high pressure systems.

For the next day, the cyclone moved little and the circulation center remained exposed to the northwest of the cloud cover. Nevertheless, it became a little more organized on August 27 and was upgraded to Tropical Storm Erin. That night, it picked up speed toward the north under the influence of an upper-level trough and passed about halfway in between Bermuda and the coast of North Carolina. Shear increased again and Erin weakened back to a tropical depression on August 28. The circulation began to lose definition too. By the morning of the 29th, Erin had started to merge with a nearby front and became post-tropical.

The above image shows Tropical Depression Six just before being upgraded to a tropical storm.

Erin did not affect any landmasses, but its remnants eventually brought heavy rainfall to Atlantic Canada.

Hurricane Dorian (2019)

Storm Active: August 24-September 7

Through mid-August, the presence of Saharan dry air in the main development region (MDR) of the Atlantic between west Africa and the Lesser Antilles stifled the tropical waves that periodically traversed the basin east to west. This finally began to change on August 23 when a low pressure system formed in association with a tropical wave a bit more than halfway to the Caribbean from Africa. Already, spin was evident on satellite imagery, but thunderstorm activity was at first quite limited. However, the low managed to battle the aforementioned dry air in its vicinity and consolidate further, becoming Tropical Depression Five late in the morning on August 24. Not long after, convection concentrated near the center of circulation and it strengthened into Tropical Storm Dorian.

At first, wind shear gave the cyclone a ragged appearance on satellite imagery and limited the upper-level outflow characteristic of a healthy tropical cyclone. This shear lessened on the 25th and Dorian took advantage, beginning to intensify slowly. Meanwhile, the system maintained its heading due west. Deep convection blossomed more consistently, keeping dry air at bay. The inner core was struggling, with multiple vortices and an ill-defined circulation center. This did not prevent it from bringing heavy rains to Barbados late on August 26, when Dorian became one of the only tropical cyclones recorded to pass directly over the island. The system also entered a weakness in the ambient subtropical ridge and turned west-northwest. Several factors favored intensification for Dorian, including low shear and warm ocean temperatures, but the circulation had dry air to contend with and could not get its center together; the mid-level and surface circulations were decoupled (displaced over 100km from one another). Further, the structure that Dorian managed to accumulate was torn apart when it passed directly over St. Lucia early on August 27 and entered the Caribbean.

The battered low-level center dissipated after contact with land and reformed farther north, resulting in a more vertically stacked circulation. This also had a significant impact on the eventual track, because it put Dorian on a northwestward heading. Late on the 27th, intensification resumed and an eyewall appeared on radar imagery. As the cyclone passed over the U.S. Virgin Islands early on August 28, a massive burst of convection appeared in the eyewall. Within hours, Dorian reached hurricane status. Some islands recorded sustained winds to near hurricane force and over 5 inches of rain. The storm's winds increased a bit further through the evening, though patches of dry air out of the south gave a few parting shots at Dorian's core as it moved northwestward into a moister atmosphere. By early on the 29th, Dorian had left the Caribbean behind.

Hints of an eye appeared on microwave satellite imagery that day, but the core only slowly improved. This was in part due to an upper-level low over the Bahamas imparting moderate shear out of the southwest. The limited shear did not prevent Dorian from reaching category 2 status that evening. For most of the day of August 30, the cyclone still had a curious “squashed” appearance, with excellent outflow channels toward the east and west, but a relatively narrow cloud shield north-to-south. The eye did begin to clear out in earnest though, and Dorian steadily intensified into the first major hurricane of the 2019 season. Simultaneously, the same upper-level low that had sheared Dorian helped it to turn back toward the west, in conjunction with a building ridge over the western subtropical Atlantic. Once the hurricane was north of the upper-level low and moving west, the flow was aligned with Dorian’s forward motion. Shear therefore diminished and left the system in near-perfect conditions for rapid intensification.

Late on August 30, it took advantage. A large, symmetric she cleared out at last on visible satellite. Dorian’s central pressure plummeted and its winds increased to match; within a few hours, it was a category 4 hurricane. The trends continued on the 31st as sustained winds reached 150 mph, high-end category 4 strength. Even colder cloud tops appeared in the now very circular storm, bringing Dorian to category 5 status early on September 1. The extraordinary bout of strengthening did not abate until that afternoon, when Dorian reached its peak intensity of 185 mph sustained winds and a central pressure of 910 mb. Only a handful of Atlantic hurricanes on record had ever achieved winds of this magnitude, and none had ever impacted the Bahamas at such an intensity. The eye of Dorian passed over Abacos Island of the northern Bahamas at peak intensity, causing catastrophic damage to the island. Even worse, steering currents collapsed that evening and the system stalled early on September 2 near the neighboring Grand Bahama. These two islands, especially the second, therefore spent many hours in the destructive eyewall of the hurricane.

Dorian’s strength finally began to ebb as the strong winds caused upwelling of cooler ocean water from beneath the surface (a common occurrence for stalled hurricanes). Late in the morning on the 2nd, it weakened to a category 4. This did little to improve the situation in Grand Bahama, however, for the eye remained stationary just north of the island that whole day. An eyewall replacement cycle occurred during the afternoon, having the net effect of enlarging wind radii but lowering the peak strength of the cyclone. Early on September 3, Dorian weakened to a category 3 and finally budged northwest as a weakness developed in the ridge. The core of the storm ingested some dry air that afternoon, though it was quickly incorporated into a ragged eye that evening. As often occurs for systems gaining latitude, Dorian continued to expand, and it passed close enough to the central Florida coastline to cause tropical storm force winds in its outer rainbands. Maximum winds dropped a little more though and the cyclone weakened to a category 2 later on the 3rd.

As Dorian moved away from its own cold wake, it produced colder cloud tops again, aided by the Gulf Stream waters. It picked up a little more speed and turned north-northwest during the morning of September 4. Fortunately for the U.S. southeast, an approaching front facilitated Dorian turning farther to the right. It traveled roughly parallel to the coastline that day, turning due north around the latitude of northern Florida and then north-northeastward as it moved toward the Carolinas. Meanwhile, its structural improvements allowed slight strengthening, and Dorian regained major hurricane strength late on the 4th, reaching 115 mph winds and a 955 mb pressure. During the morning of September 5, the western eyewall swept over coastal South Carolina and it weakened back to category 2 strength. The close brush caused significant storm surge inundation and hurricane force wind gusts. Overnight, Dorian's center made an even closer approach to North Carolina. The winds decreased to category 1 strength, but the central pressure remained steady just below 960 mb, and the inner core structure was remarkably stable. Late in the morning on September 6, the cyclone made an official landfall in Cape Hatteras, North Carolina as it accelerated northeastward. Even as it gained latitude, Dorian continued to generate new deep convection in the eyewall through that night.

Finally, early on September 7, the cyclone began to lose its warm core and extratropical transition began over the cold waters north of the Gulf Stream. Even as Dorian became asymmetric, it actually deepened and intensified. This was due to baroclinic processes by which extratropical cyclones intensify, such as large temperature gradients in the atmosphere along frontal boundaries. Dorian regained category 2 status that morning, peaking again at 100 mph winds and a pressure of 953 mb southwest of Nova Scotia. Hurricane force winds and heavy rain caused widespread damage and loss of power in Atlantic Canada. During the afternoon, Dorian at last became post-tropical just before making landfall near Halifax, Nova Scotia. Its effects continued overnight though as it moved across the Gulf of St. Lawrence toward Newfoundland. The extratropical cyclone's winds dropped below hurricane strength during the afternoon of September 8 before the center passed over northwestern Newfoundland. During the night, ex-Dorian finally moved away from land over the far northern Atlantic. It moved east-northeastward for a few days before dissipation.

The above shows Dorian at peak intensity over the Bahamas. It was the strongest Atlantic hurricane ever recorded that far north (at 26.6 °N).

The above shows the long track of Dorian. Its worst impacts were in the Bahamas, the coastal Carolinas, and Atlantic Canada (as a post-tropical storm).

Tropical Storm Chantal (2019)

Storm Active: August 20-23

In the middle of August, a frontal system moving toward the U.S. southeast coast began to stir up thunderstorm activity over the Carolinas and northeastern Florida. Near its southern end, a trough developed, and on August 17 a low pressure center. It moved steadily northeastward along the Carolina coastline and then out over the open ocean. Conditions were not very favorable but nevertheless a small area of thunderstorm activity persisted near the center. On August 19, the disturbance turned due east, traveling out to sea at a fast clip. The next day, the circulation became well-defined enough for the system to unexpectedly develop into Tropical Storm Chantal.

As the tropical storm moved slightly south of east, sea temperatures actually warmed modestly and wind shear decreased. However, these favorable factors were balanced by a major inhibitor: the atmosphere was very dry over the open northern Atlantic. As a result, Chantal struggled to maintain its convection and gradually weakened, becoming a tropical depression by August 22. The gradual spin-down continued the next day and the cyclone degenerated into a remnant low late on August 23. By this time, the low had turned southeast and slowed down. It dissipated soon after.

The above image shows Chantal moving eastward away from the North American coastline.

Chantal was a short-lived and weak tropical storm which did not affect land during its time as a tropical cyclone.

Tropical Depression Three (2019)

Storm Active: July 22-23

Around July 12, a tropical wave moved off the coast of Africa. It was among the first of the season to be seriously monitored for cyclone development, but it traversed the Atlantic basin for the following week without incident. A portion of the wave axis took a northern route, passing north of the Caribbean islands and approaching the Bahamas by July 21. Stable dry air in the region made progress difficult for the disturbance, but it managed to spin up a small area of convection driven by very warm ocean waters. This led to a tiny circulation and the system strengthened into Tropical Depression Three on July 22 over the western Bahamas.

Soon after, the depression began to feel the influence of an approaching cold front and turned northward on July 23. The center passed just offshore of east Florida, but its small size meant that only a few showers and occasional gusty winds impacted land. By the late morning, the system had already lost its identity and dissipated as it combined with the front.

Even though the tropical depression formed over very warm water, it succumbed quickly to dry mid-level air.

Tropical Depression Three was a small and short-lived system with minimal land impacts.

Hurricane Barry (2019)

Storm Active: July 11-14

During the second week of July, a trough of over the southeast United States drifted slowly south-southeastward, producing some scattered afternoon thunderstorms as it went. A few days later, on July 9, this anomalous motion brought it into the extreme northeastern Gulf of Mexico, where more consistent convection began to flare up. Weak steering currents allowed the system to meander west-southwestward and it gradually organized, developing a broad circulation. By July 10, a clear low-level center had formed, but it was displaced well to the northeast of the mid-level circulation. Moreover, the strongest thunderstorms were actually located over southeast Louisiana, where significant flooding occurred before the tropical disturbance had even been classified.

Finally, on July 11, improvements in organization prompted the naming of Tropical Storm Barry, located now nearly due south of the Mississippi delta. Even after naming, however, dry continental air pushing in from the other restricted cloud cover to the southern portion of the tropical storm, and several small low-level vortices were evident on satellite imagery. This disorganized hampered Barry's intensification. Nevertheless, the pressure fell appreciably over the next day and aircraft reconnaissance indicated that the system's maximum winds steadily increased to strong tropical storm strength by July 12.

Meanwhile, Barry took a turn north of west around the edge of the mid-level steering ridge and began to move toward Louisiana. Even by the afternoon of July 12, however, the northern semicircle remained very dry, so few effects were felt over land even with the system less than 100 miles offshore. Despite its unconventional structure, Barry steadily strengthened through landfall. It peaked as a category 1 hurricane with 75 mph winds and a pressure of 993 mb on July 13 as it crossed the central Louisiana coastline around noon local time. The slow movement of the system resulted in only a gradual weakening trend and prolonged heavy rainfall, especially just east of the landfall point. Nevertheless, Barry weakened to a tropical storm shortly after landfall and a tropical depression on July 14 as the center of circulation pushed further inland. Upper level winds out of the north kept most of the precipitation over water even as the center moved away, sparing inland areas from more severe flooding. Soon after, the storm became extratropical over the midwest.

The above image shows Hurricane Barry near landfall, with most of the northern half of the circulation exposed.

Barry originated from a non-tropical disturbance over the southeast U.S.

Subtropical Storm Andrea (2019)

Storm Active: May 20-21

As the third week of May began, a frontal boundary moved off of the U.S. east coast. The southern end of the front stalled north of Hispaniola and formed a trough of low pressure. There the system found a relatively favorable atmosphere and marginally warm ocean temperatures, supporting some scattered storm development. Before long, a low-pressure center had developed. By May 20, the storm had a small convective shield displaced to the north and east and aircraft reconnaissance measured gale-force winds. Since the circulation was still interacting with an upper-level low to its southwest and the gale force winds were spread out from the center, the storm was classified Subtropical Storm Andrea, the first named storm of the 2019 Atlantic hurricane season.

The system moved northward that day, but began to slow down and veer eastward by the the afternoon of May 21. Meanwhile, the convection associated with the system dissipated, leaving behind just a swirl of low-level clouds to mark the center of circulation. As a result, Andrea was downgraded to a subtropical depression. Early that evening, it degenerated further into a remnant low and these remnants dissipated the next day as a new front approached. Moisture that had been associated with Andrea brought some rain showers to Bermuda on the 22nd. The formation of Andrea marked the 5th consecutive year during which a named storm formed prior to the official start of the hurricane season on June 1, surpassing the record set in 1951-4. However, short-lived weak systems such as Andrea may very well have been missed prior to the era of satellite observation.

This image shows Subtropical Storm Andrea on May 20. Also visible is the upper-level low to its southwest which helped to weaken the system.

Andrea formed in the far western Atlantic, one of the typical areas for early season cyclogenesis.

Professor Quibb's Picks – 2019

My personal prediction for the 2019 North Atlantic hurricane season (written May 19, 2019) is as follows:

15 cyclones attaining tropical depression status,
14 cyclones attaining tropical storm status,
6 cyclones attaining hurricane status, and
3 cyclones attaining major hurricane status.

Following a fairly average hurricane season in 2018 (which nevertheless featured two devastating major hurricanes), I predict that the 2019 season will see a comparable number of cyclones, albeit with rather different areas to watch. Note that the average Atlantic hurricane season (1981-2010 average) has 12.1 tropical storms, 6.4 hurricanes, and 2.7 major hurricanes. As with any season, our prediction begins with a look at the El Niño Southern Oscillation (ENSO) index, a measure of equatorial sea temperature anomalies in the Pacific ocean that have a well-documented impact on Atlantic hurricane activity. These anomalies are currently positive, corresponding to an El Niño state, and have been since last fall. The image below (click to enlarge) shows model predictions for the ENSO index through the remainder of 2019.

In comparison to the last several years, the situation is more static: no significant change of state is expected during this year's hurricane season (though there is, of course, significant uncertainty). This state of affairs tends to suppress hurricane activity and increase the chance of cyclones in the subtropical Atlantic curving away from the North American coastline (unlike, for example, the unusual track of Hurricane Florence last year).

This is fortunate, because all indications are the subtropical Atlantic will continue to churn out named storms as it did last season. Sea surface temperatures continue to run high in the region, and El Niño effects are not as pronounced there, partially explaining why my prediction still features an above average number of storms. Other factors also somewhat offset the El Niño: ocean temperatures in the tropical Atlantic (the birthplace of most long-track hurricanes) are slightly above normal this year, a trend expected to persist over the next several months. The atmosphere has also been less dry in the region, with less Saharan dry air than in 2018 and the beginning of the 2017 season to quash developing tropical waves. Expect the tropics to be less hostile to long-track hurricane formation than last year, when all cyclones taking the southerly route dissipated upon entering the Caribbean.

My estimated risks on a scale from 1 (least risk) to 5 (most risk) for different specific parts of the Atlantic are as follows:

U.S. East Coast: 3
Though the subtropical Atlantic will be active, I predict less of a risk to the U.S. coastline, with a smaller chance of a Florence-like system this year. Though there may be a few hurricanes passing offshore, most should recurve out over open water. Bermuda, however, is at higher risk.

Yucatan Peninsula and Central America: 2
These regions may benefit the most from a persistent El Niño, with wind shear making the development of an intense hurricane in the western Caribbean difficult. Further, I expect tracks to curve northward more often than striking Central America directly. Later season cyclones originating in the monsoonal gyre near Panama may pose the primary threat, and these tend to be principally rainmakers.

Caribbean Islands: 4
With the main development region (MDR) of the tropical Atlantic more favorable this year, the Caribbean is unlikely to continue the reprieve last year that followed arguably its worst season of all time (2017). Early season storms are still likely to fizzle out due to El Niño-related shear, but a wetter atmosphere suggests that tropical disturbances will have to be watched carefully. This includes a greater possibility of tropical cyclogensis in the Caribbean itself.

Gulf of Mexico: 3
Sea temperatures are consistently higher in the Gulf this year than they have been recently, especially near the Florida gulf coastline, but conditions here overall are a mixed bag. A strong jet stream across the continental U.S. will support more severe thunderstorms over land this summer, but this actually may work against cyclones thriving in the region. Balancing these factors yields an average risk, though this overall rating is a combination of a higher-than-normal risk in the eastern Gulf and a lower-than-normal risk farther west.

Overall, I expect the 2019 hurricane season to feature close-to-average activity. Nevertheless, this is just an informal forecast. Individuals in hurricane-prone areas should always have emergency measures in place. For more on hurricane safety sources, see here. Remember, devastating storms can occur even in otherwise quiet seasons.

Sources: https://www.cpc.ncep.noaa.gov/products/analysis_monitoring/lanina/enso_evolution-status-fcsts-web.pdf, https://www.cpc.ncep.noaa.gov/products/CFSv2/CFSv2seasonal.shtml, https://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

Hurricane Names List – 2019

The name list for tropical cyclones forming in the North Atlantic basin for the year 2019 is as follows:

Andrea (used)
Barry (used)
Chantal (used)
Dorian (used)
Erin (used)
Fernand (used)
Gabrielle (used)
Humberto (used)
Imelda (used)
Jerry (used)
Karen (used)
Lorenzo (used)
Melissa (used)
Nestor (used)
Olga (used)
Pablo (used)
Rebekah (used)
Sebastien (used)
Tanya
Van
Wendy

This list is the same as the list for the 2013 season, with the exception of Imelda, which replaced the retired name Ingrid.

The abc Conjecture: Applications and Significance

This is the third part of a three-part post concerning the abc conjecture. For the first, see here.

The first post in this series presented some explanation as to why the abc conjecture seems like a reasonable attempt to mathematically codify a big idea. This idea is that the prime factorization of a sum of two numbers should not really relate to those of the individual numbers. Equivalently, it says that if we see an equation like 3 + 5³ = 2⁷, we should think of it as a "rare event" or "coincidence" that big powers of small primes are related in this way. The second post provided some examples and numerical evidence rigorous version of the conjecture. To review, this states that

The abc Conjecture: For any ε > 0, no matter how small, for all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc)^{1 + ε} > c.

Again, the radical rad(n) of an integer n is the product of its distinct prime factors. However, none of what has been discussed so far constitutes a mathematical proof that the abc conjecture is true or false.

In 2012, the Japanese mathematician Shinichi Mochizuki shocked the mathematical community by publishing, out of the blue, what he claimed was a proof of the abc conjecture. However, the initial excitement at this announcement was quickly replaced by confusion; almost no one was able to decipher the tools used in the proof, which totaled over 500 pages in length! Mochizuki, working in isolation for years, had built up a brand new mathematical formalism which he called "Inter-Universal Teichmüller Theory" that was bizarre and unfamiliar to other researchers. The language and notation (an sample of which is provided in the screenshot below) seemed alien, even to mathematicians!

Moreover, he refused to publicly lecture on the new material, instead only working with a few close colleagues. The combination of the length and inscrutability of the proof with his unwillingness to elucidate it discouraged people from attempting to understand it. In the years since the proof was published, skepticism has mounted concerning the proof's validity. While a small group of mathematicians defend it, a majority of the mathematical community thinks it is unlikely that the proof is valid. For now, the abc conjecture remains effectively open.

Nevertheless, it is certain that attempts to prove the conjecture will continue. It has a number of useful applications that would solve a myriad of other mathematical problems, should it be true. To illustrate the power of the abc conjecture, we give one famous example of an application: Fermat's Last Theorem.

One of the first equations we considered in this series was x² + y² = z², which relates the side lengths of right triangles. This equation has infinitely many solutions, namely 3² + 4² = 5², 5² + 12² = 13², etc. Fermat's Last Theorem states that if we raise the exponents from 2 to any higher power, there are no solutions in the positive integers. That is, x³ + y³ = z³, x⁴ + y⁴ = z⁴, and so on are not satisfied by any x, y, and z > 0. Famously claimed by Pierre de Fermat in the 17th century, this problem remained unsolved for centuries. In 1985, when the abc conjecture was first stated, it remained open.

So let us assume that we have (somehow) proven the abc conjecture, and were interested in Fermat's Last Theorem. The first thing to note about the equation xⁿ + yⁿ = zⁿ is that if we had a solution for this equation, we could always find one for which xⁿ and yⁿ were relatively prime. This is because if they have a common prime factor, so must zⁿ, and we can cancel this factor (raised to the nth power) from both sides. Therefore, we have arrived at a situation in which we can apply the abc conjecture. The radical of xⁿ, for any n, is at most x since multiplying x by itself does not introduce any more prime factors that were not already there. Hence rad(xⁿyⁿzⁿ) = rad(xⁿ)rad(yⁿ)rad(zⁿ) ≤ xyz < z³. Therefore, for ε > 0, we have that

rad(xⁿyⁿzⁿ)^{1 + ε} < (z³)^{1 + ε} = z^{3 + 3ε}.

On the other hand, applying the conjecture to this triple, we have that for ε > 0,

rad(xⁿyⁿzⁿ)^{1 + ε} > zⁿ

in all but finitely many cases. Since we can choose ε to be any positive number, we can make it small enough so that 3 + 3ε < 4 (e.g. if ε = 0.1). Then if n ≥ 4, the two inequalities above directly contradict each other. Since the top one always holds and the bottom holds in all but finitely many cases, we conclude that there can be at most finitely many exceptions to Fermat's Last Theorem when n ≥ 4.

So the abc conjecture does not quite imply Fermat's Last Theorem, but it comes very close. If, in addition, we knew just a bit more about how the exceptional abc triples behaved, we could manually verify that there are no counterexamples to Fermat's Last Theorem for n ≥ 4. Interestingly, this argument does not say anything about the n = 3 case, that is, about the non-existence of solutions to x³ + y³ = z³. This special case, however, had already been proven by Euler in the mid-1700s.

Of course, the abc conjecture remains unproven, while Fermat's Last Theorem was finally proven by Andrew Wiles in 1995. This was done by entirely different means. Nevertheless, this serves as a relatively simple example of how the conjecture can prove results about Diophantine equations without invoking very difficult mathematics. Another example of a consequence is the following statement, sometimes called Pillai's conjecture:

Conjecture: Every natural number k occurs only finitely many times as the difference of two perfect powers.

For example, the special case k = 1 is the subject of Catalan's conjecture, and states that x^p - y^q = 1 has only one solution: 3² - 2³ = 1. This was proven by Preda Mihăilescu in 2002 (again by very different means from those above and from Wiles' methods), but the general case remains unsolved. If we knew for a fact that the abc conjecture were true, we would be able to prove this result by a very similar argument to the one given above for Fermat's Last Theorem (the reader is encouraged to try this!). Note that Pillai's conjecture also implies that the original equation that motivated the abc conjecture, namely y² = x³ + k, also has only finitely many solutions (for fixed k). This is the result David Masser and Joseph Oesterlé sought on their way to first formulating the statement.

These examples start to indicate how important the abc conjecture is to the study of Diophantine equations; if it were proven, it would resolve many different problems that are currently treated separately in a single stroke. Even reproving known results in a new and simple way would be greatly beneficial to the theory, since a set of tools that could prove abc would help to unify disparate parts of number theory. As a result, mathematicians will doubtlessly continue work toward solving the conjecture and probing the most fundamental structure of numbers.

Sources: http://projectwordsworth.com/the-paradox-of-the-proof/, Shinichi Mochizuki: Inter-Universal Teichmüller Theory I: Construction of Hodge Theaters, http://mathworld.wolfram.com/PillaisConjecture.html, https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The abc Conjecture, 12 sep 2013.

The abc Conjecture: abc Triples

This is the second post in a series about the abc conjecture. For the first post, see here.

In the last post, we defined the radical of an integer n, namely the product of distinct prime factors of n. We suspected in the last post that for most equations a + b = c where a and b are relatively prime, rad(abc) > c. This is because this inequality expresses our hypothesis that there should not be too many high powers of primes in the factorizations of a, b, and c. As a result, we made the following conjecture:

Conjecture 1: For all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc) > c.

However, as mentioned at the end of the previous post, this is in fact false. To prove this, we have to exhibit an infinite family of equations a + b = c with rad(abc) ≤ c. Any triple (a,b,c) of numbers satisfying this property is called an abc triple. The only example we've seen so far is (1,8,9), or in equation form, 1 + 8 = 9. In terms of this new definition, we are trying to show that there are infinitely many abc triples. The following claim gives the desired result.

Claim: For any prime number p grater than 2, the triple (a,b,c) = (1,2^p(p-1) - 1,2^p(p-1)) is an abc triple.

Proof: This family is infinite because there are infinitely many prime numbers p. The proof depends on a fact in elementary number theory known as the Euler-Fermat Theorem. This theorem may be used to show that b = 2^p(p-1) - 1 is divisible by p². This is significant because we now know that the radical of b cannot be greater than b divided by p; this is because taking the radical of b "forgets" about at least one of the factors of p. Of course, rad(1) = 1 and rad(2^p(p-1)) = 2 so

Since p > 2, this last value is less than c, so that we do in fact have an infinite family of abc triples.

In fact the situation is even worse than this. Since the radical is less than 2c/p (as shown in the proof), it is not enough to replace the hypothesis rad(abc) > c with 2rad(abc) > c, or any higher multiple. We can make 2/p arbitrarily small by increasing p so that the radical is smaller than c by an arbitrarily large factor. For example, taking p = 5 gives the abc triple (1,1048575,1048576). Note that 5² = 25 divides b = 1048575, as claimed. Our proof guarantees that rad(abc) ≤ 2c/5. In fact the radical of this product is 419430. This is indeed less than 2/5 of c.

All of this shows that we cannot correct our conjecture 1 by adding a multiplicative factor to our inequality. The next reasonable thing one might try is a power law. Perhaps rad(abc)² > c for all but finitely many equations, or something similar. This, in fact, is the correct idea. However, the choice of 2 as the exponent again seems arbitrary. We know already that the statement is false when the power is 1, so let's try increasing it just a little. This leads us to the actual abc conjecture.

The abc Conjecture: For any ε > 0, no matter how small, for all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc)^{1 + ε} > c.

The variable ε could for example be 1, and then we recover the rad(abc)² > c inequality. However, ε could also be very close to 0, giving an exponent of 1 + ε very close to 1. Crucially, any function x^{1 + ε} with ε > 0 eventually increases faster than any constant multiple of x, for example x^1.1, x^1.00001, etc. Therefore, this conjecture gets around the counterexample to conjecture 1. Nevertheless, the abc conjecture in some sense says that conjecture 1 is really close to being true. All we needed to do was increase the exponent by any positive amount. These concepts may become a little clearer with a new concept, called the quality of a triple (a,b,c). The formula for the quality, denoted q(a,b,c) is

This is another measure of how large c is compared to rad(a,b,c). In fact, rad(a,b,c)^q(a,b,c) = c. For example, q(13,22,35) is about 0.386, and q(1,8,9) is close to 1.226. This allows a more succinct description of the conjecture: for most triples, q ≤ 1. It follows from our definition that abc triples are those for which q > 1. Finally, the abc conjecture is equivalent to the following.

The abc Conjecture (Second Formulation): For any ε > 0, no matter how small, for all but finitely many equations of the form a + b = c where a and b are relatively prime, q(a,b,c) < 1 + ε.

Let's see if our conjecture seems plausible from the numerical data. One possible way to do this is to come up with many triples and see how large the quality q is for each.

In the above diagram (click to enlarge), the x-axis is our variable c. For each c between 2 and 2000, the plot goes through all possible relatively prime values a and b adding to c, finds the triple among these with the highest quality, and plots a corresponding point there. Therefore, all points are already among the highest quality triples. Even among these, abc triples (those that lie above the horizontal q = 1 line) are rare. Furthermore, they seem to get even more rare as c increases. In terms of diagrams of this sort, the conjecture states that only finitely many dots lie above a given horizontal line q = 1 + ε for any ε > 0. The highest quality abc triple that appears on the plot is (3,125,128) = (3,5³,2⁷), with a quality of 1.426. Are there any higher quality triples out there?

In fact, there are well over a hundred known with higher quality, a list of which may be found here. Currently, the highest known quality belongs to the triple (2,6436341,6436343) = (2,3¹⁰109,23⁵), with q = 1.6299. Even assuming the abc conjecture does not answer the question of whether this triple is really the highest quality there is. All it says is that examples of this sort must eventually die out as we approach infinity. For instance, there may very well be no triples at all with q ≥ 2, meaning that c ≤ rad(abc)² may hold in all cases with no exceptions.

Of course, no matter how many examples we check, we are no closer to proving that the abc conjecture holds. In the last post, we will discuss attempts to prove it, as well as the applications of the statement, should it be true.

Sources: http://www.math.leidenuniv.nl/~desmit/abc/, Greg Martin and Winnie Miao: abc Triples; Arxiv:1409.2974v1 [math.NT] 10 sep 2014, Brian Conrad: The abc Conjecture, 12 sep 2013.