Fermat's Last Theorem - A Primer

Fermat's Enigma, written by Simon Singh, is a book about Fermat's Last Theorem, its origin, history and how it was officially proved in 1997. One day Lily picked up this book for kids' non-fiction reading. I scaned a few pages and could not put it down - I was fascinated.

Fermat's Last Theorem

The theorem was in fact a conjectecture, first proposed by french mathematician Pierre de Fermat, around 1637. It was based on the Pythagorean Theorem:

In a right hand triangle, the square of the hypotenuse (z) is equal to the sum of the square of the other tow sides, i..e. x^2+y^2=z^2.

A peculiar feature of the equation is that it has infinite whole number (integer) solutions, e.g. (3,4,5), (5, 12, 13), .....

It is noted that Chinese and Babylonians discovered the relation one thousand years before Pythogoras. However it was mathematically proved by Pyththagoras of Samos, an ancient greek mathmatician (~ 600BC), and thus the name after him.

Fermat's last theorem
The following equation

x^n+y^n=z^n (*)

has no integer solution if n>2.

It is intriguing that equation (*) has infinite number of integer solutions for n=2, and has no integer solutions at all for n>2. For 350 years since the proposal of the theorem no body can prove or disapprove the theorem.

People behind the Math

The math involved in the history of proving the Fermat's last theorem is interesting. The life stories of the involved mathematicians are fascinating - their life, their work, their failures in proof, triumphs in reaching major milestones. Five mathematicians were of great significance to the Fermat's last theorem and its proof: Pythagoras(600BC), whose Pythagorean theorem inspired P. Fermat to make the conjecture; Fermat, who intrigued and frustrated mathematicians centuries to follow with his notes on the margin of a mathematics book(1637); L. Euler, the 18th century genius, who made the first breakthrough in proving Fermat's last theorem (1753), 100 years after Fermat's death; P. Wolfskehl, who helped to sustain interests in the proof of the theorem by setting up a Wolfskehl Prize in 1908 and Andrew Wiles, the Princeton professor who proved Fermat's last theorem in 1997.

Fermat, who was actually a judge, was brilliant and smarter than most of his contemporary mathematicians. He was into math because it was fun for him. He did not publish any of his work because he did not like to provide full proof of the problems he solved. He usually just wrote down an outline of how he solved the problem. Of course the most famous such sketch is the note he left on the margins of the book II of Arithmetica. He stated his conjecture there and then he continued to write " ...I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain". However he did enjoying challenging others by posting problems he solved. He had particular pleasures to see others struggle on those problems! Thanks to Fermat's son Clement-Samuel, who collected his father's notes and published them in the book Arithmetica Containing Observations by P. de Fermat, we know Fermat's remarkable work in number theory, and the ensuing centuries of struggle to prove his last theorem.

Pythagoras of Samos, an ancient Greek mathematician (~ 600BC) traveled the world to learn math, especially from Egyptians and Babylonians. He laid the very foundation of mathematical proof on top of Pythagorean theorem. A very interesting story was that he paid his first student three silver coins for each lesson taken. After sometime, the boy became enthusiastic about mathematics. Pythagoras then stopped paying his student, pretending that he did not have any more money. The student offered to pay to continue to receive education from Pythagoras.

To advance mathematics, he established Pythagoras Brotherhood, a cult type of mathematical school. He believed everything was a rational number - the whole numbers and fractions. He took this belief so sacredly that it prevented him from accepting the existence of irrational numbers, such as square root of 2. A student of Pythagoras, Hippausas attempted to find a fraction for square root of 2, and eventually he proved that no such fraction existed. Pythagoras did not want to admit the existence of irrational number, but he could not deny its existence by mathematical logic. Tragically he resorted to force, sentencing the student to death by drowning!!

Leonhard Euler, the great mathematician from Swiss, worked in Russia most of his life. Contrary to Fermat, Euler liked to write papers. He proved the theorem for n=3 using imaginary number to plug holes in his proof. His attempts to extend the proof all ended in failure. Euler showed exceptional talent in math at young age but his father wanted him to study theology, and he dutifully obeyed. Luckily the Bernoulli family - one of the Bernoulli was responsible for the Bernoulli principal in fluid mechanics, was in the same town Euler lived and two of the Bernoullis were Euler's good friends. The Bernoullis appealed to Euler Sr. to let Euler pursue a career in math. Because Euler Sr. had great respect to the Bernoullis, he reluctantly gave in. The world got one of its greatest mathematicians.

Paul Wolfskehl was an average mathematician from a wealthy family. He created Wolfskehl Prize due to an aborted suicide. The story goes like this. A failure in his pursuit of a beautiful woman led to his plan to kill himself. He set a date and would kill himself at exactly midnight that day. He was so efficient in carrying out his plan, he finished the preparation for his suicide attempt ahead of time. So he went to the library to scan through mathematical publications and he happened to read a paper on why two major efforts to prove Fermat's last theorem by Lame and Cauchy respectively failed. He was quickly absorbed into the paper and found an error in it. By dawn he remedied the error in the paper, and his despair evaporated. He tore his will and wrote a new one to create a prize for the proof of Fermat's last theorem.

Andrew Wiles, the Princeton math professor, who worked in secret for 7 years during his pursuit of the proof, has a life long obsession with Fermat's last theorem starting at 10 after he read a history book on math. After he announced that he had proved Fermat's last theorem at a math conference, his proof was subject to peer review. Referees found a potential crushing error in the proof. Wiles sweat for about a year and with substantial help from others, finally fixed the error in the proof. The proof was eventually published in the journal Annals of Mathematics in 1995. The Wolfskehl Prize was presented to him in 1997.

(to be continued ... Numbers)

Our Yellowstone Trip Plan

Trip plan helps us to enjoy a trip. As for any major trips we have had, I prepared a detailed trip plan for our Grand Teton/Yellowstone trip.

The first two days and the last day of the trip we stayed at Jakson Hole. During the Yellowstone portion, we stayed in a hotel at West Yellowstone, a small town just outside the west enstrance.

Day 1: Flight from DFW to Jackson Hole arriving at noon. (Flight was delayed due to mechanical problem, afertnoon activity canceled, rescheduled to the end of the trip)

Afternoon activity – Ride gondola and hiking at Jackson Hole Mountain Resort

Day 2: Grand Teton – whole day

Morning: Scenic Drive - Jenny Lake Scenic Drive, Hike to Hidden Falls/Inspiration point

Afternoon – white water rafting

http://www.davehansenwhitewater.com/whitewater.shtml#8manraft

Day 3: Yellow Stone National Park – day 1 - Geysers

Morning - stop at Ox Bend before entering Yellowstone from south entance

1) Experience Old Faithful, the most popular geyser in the world, and hundreds of other geysers and hot springs

2) Kepler cascade: Kepler Cascades is the most easily reached waterfall in the district. A marked pullout just south of Old Faithful and a short walk from the car offers the visitor easy access to view this 125-foot cascade.

3) Mystic Falls Trail
This trail follows a lovely creek through a lodgepole pine forest before reaching the 70- foot falls. By following a series of switchbacks, an overlook of the Upper Geyser Basin can be reached before looping back to join the main trail.

4) Go to West Yellowstone for lodging

Day 4: Yellow Stone National Park – day 2 - Canyon Village

1) On the way – stop at Madison, Artist Paint Pots

2) Gibbon Falls

This 84-foot (26-meter) waterfall tumbles over remnants of the Yellowstone Caldera rim. The rock wall on the opposite side of the road from the waterfall is the inner rim of the caldera.

3)Norris Geyser Basin is the hottest, oldest, and most dynamic of Yellowstone's thermal areas.

4) Grand Canyon of Yellow Stone

Day 5: Yellowstone National Park – day 3 - Mammoth hot spring

On the way - Roaring Mountain

1) The Gardner River and Gardner River Canyon
The North Entrance Road from Gardiner, Montana, to Mammoth Hot Springs
2) Mammoth Hot Springs
3) Beavers Pond Trail
4) Ranger's program
5) Lemar Valley

Day 6: Yellowstone National Park – day 4- Tower-Roosevelt

To via Mammoth, back via Canyon village

1) Petrified Tree Specimen Ridge
Specimen Ridge, located along the Northeast Entrance Road east of Tower Junction, contains the largest concentration of petrified trees in the world. There are also excellent samples of petrified leaf impressions, conifer needles, and microscopic pollen from numerous species no longer growing in the park. Specimen Ridge provides a superb "window" into the distant past when plant communities and climatic conditions were much different than today.

2) Tower Fall
Tower Fall is the most recognizable natural feature in the district. The 132-foot drop of Tower Creek, framed by eroded volcanic pinnacles has been documented by park visitors from the earliest trips of Europeans into the Yellowstone region. Its idyllic setting has inspired numerous artists, including Thomas Moran. His painting of Tower Fall played a crucial role in the establishment of Yellowstone National Park in 1872.

3) Natural Bridge Trail
The natural bridge is a 51 ft. (18 m) high cliff of rhyolite rock that has been cut through by the erosional forces of Bridge Creek. The trail from the campground meanders through the forest for 1.2 mile (0.8 km). It then joins the road and continues to the right (west) for 1 mile (1.5 km) before reaching the Natural Bridge. The short but steep switchback trail to the top of the bridge starts in front of the interpretive exhibit. To protect this fragile resource, the top of the bridge is closed to hiking. However, good views may be attained next to the bridge. The bicycle trail to the bridge begins just south of the marina off the main road.

Day 7: Yellowstone National Park – day 5: Yellow Stone Lake – West thumb area

Go there via old faithful

1) Craig Pass
Craig Pass, at 8,262 feet on the Continental Divide, is about eight miles east of Old Faithful on the Grand Loop Road. In 1891, road engineer Captain Hiram Chittenden discovered Craig Pass while he was surveying for the first road between Old Faithful and West Thumb. It was probably Chittenden who named the pass for Ida M. Craig (Wilcox), "the first tourist to cross the pass" on Chittenden's new road, on about September 10, 1891. At the time that her name was given to the pass, Ida Wilcox (1847-1930) had been married 24 years. So why did Chittenden use her maiden name? Perhaps it was to honor her singularly for being the first tourist to cross the pass. It is also possible that through his connection with the military, Chittenden knew her father (Gen. James Craig) or her brother (Malin Craig, Sr.) and was really honoring the Craig family.

2) Isa Lake
Hiram Chittenden of the U.S. Army Corps of Engineers claimed to have discovered this lake on the Continental Divide at Craig Pass in 1891. Isa Lake is noteworthy as probably the only lake on earth that drains naturally to two oceans backwards, the east side draining to the Pacific and the west side to the Atlantic.

3) Boating/Kayaking on Yellowstone lake – time permitting

4) Cascade Lake Trail This hike takes 3 hours and is an enjoyable walk through open meadows and over small creeks for those with limited time. Look for wildlife and wildflowers in season. Most years, this trail remains very wet and muddy through July. Trailhead: Cascade Lake Picnic Area, 1.5 miles north of Canyon Jct. on the Tower-Canyon Road. Distance: 4.5 miles (7.2 km) roundtrip Level of Difficulty: Easy

Day 8: Return to Jackson hole

1) Kayaking @ Jenny Lake, Grand Teton NP in the moring
2) Visiting Jakson hole ski mountains in the afternoon

Day 9: Return to DFW

Note:
1) We chose the hotel outside the park for two reasons - first hard to resever lodging inside the park, and inside lodging condition was not good; second we did not want to move daily from one place to another inside the park - which would take a lot of time. Hindsight - staying outside made us drive more inside the park and thus more chances to see wildlife
2) best wildlife sighting time is early morning and early evening
3) got to have good binoculars and high zoom camera to view and capture wildlife far awy from viewing locations
4) plan a trip to Yellowstone early of the year, book hotel and plane ticket early

Wildlife We Saw at Yellowstone

(Note: Double click small size pictures, the full size photos will appear for better viewing of the far away animals)

During our trip at Yellowstone, we stayed outside the park at a hotel in the town of West Yellowstone - very close to the park's west entrance. It worked out pretty well as far as wildlife viewing and traffic was concerned. Getting in early in the morning (7am or so), leaving a bit late (~ 7pm) helped us to avoid traffic jams. Even though we drove about 30 miles of the same sections of roads every day, we actually had more chances for wildlife sightings. This might be one case that doing the same thing again and again we could expect different outcome.

The first day we entered the park from the west entrance in early morning, ~ 7am. About 10 miles into the park, we saw cars lined up by the side of road ahead of us. "Animals!" As we drew closer, we saw a large group of deers on the hill slope by the road. Despite the frequent sighting of deers, we were still very excited since this was our first animal sighting at Yellowstone. As we were about to leave the area, one deer came down the hill and crossed the road in front of our car!

One day on the way out of the park, we saw another line up of cars by roadside, we slowed down, and there it was - a red fox by the road side. We parked on the other side of the road, and observed and video taped the fox observing passers by, stretching its front legs, yawning and simply enjoying the late afternoon Sunshine.

Another day as we were on the road from one major area to another at noon time, we saw a lot of people standing by road side looking at an area with a few huge fallen trees. We were told by other onlookers that there was a male adult moose under a big tree trunk. It took us quite a bit of time to locate the moose which has a big antler. Someone must have really sharp eyes and excellent sense of moose to spot the good camouflaged animal under the fallen trees.

One time, there was an eagle fishing in a river, frequently resting on a log. A ranger set up a telescope aiming at the log, so people who did not have good binoculars could see it. I took a picture of the eagle through the lens of the telescope!

It is great to see wildlife when someone else found it. It was much better and way more exciting and thrilling if we were the first one to spot a hardly found animal, such as the grizzly bear and the trumpeter swan. The key to wildlife viewing in such an large area is to be vigilant, to be alert and to be on the look out.

The mammoth spring was a big disappointment of our Yellowstone trip because most springs were dormant that year. We went to a ranger educational program about the Yellowstone ecosystem there- which was interesting. At the end of the program, Lily asked the ranger about the best area to view gray wolf - a unique and controversial animal to the area. It was Lemar Valley - a 30 ~ 40 mile stretch to the east entrance of the park. We drove to the valley later that day. When we got there, there was not much to see - brown grass at the bottom of the valley, low grayish green bush buffered between the grass and pine trees further away, all the same along the stretch. We did see a couple of pick-ups parking along the stretch, old people with binoculars in hand waiting for gray wolfs to show up - as the ranger told us. It was getting late, we turned around, and drove through the valley again, around 6:10pm (per time stamp on pictures), Lily spotted something moving along the seam between grass and bush, "Wolfs?" "A pack of wolfs!" Justin confirmed via his binoculars. The three wolfs were walking along the seam in a line. How exciting! We were exuberant. We saw a couple of bighorn antelopes in the same area, wishing to see wolfs to prey on them. Unfortunately, the wolfs were going in the opposite direction. We watched the wolfs until they disappeared into the bushes.

Be observant, and be patient would go a long way as well.

One time we saw a group of pelicans swimming in a pond, a minute later a bison walked slowly into the picture. Another time we were witnessing a bald eagle circulating over a hill, a few minutes later, a group of white birds started following the bald eagle! Yet another time, We observed a few ospreys over a pond. Then one osprey starting to flap its wings, stayed stationary in the air, looking down, a few seconds later it dived into the pond - apparently to catch fishes. It was so fast, and so far away, it was really difficult to capture the actions on camera or video though we observed the dive several times.


Timing is hugely important as well.

In our second day at Yellowstone, we hiked along the beautiful and scene ever changing beavers pond trail. We saw a pheasant on the trail, a few seconds later, it disappeared into the woods. When reached the beavers pond, we saw a beaver dam and ducks in the pond but beavers were nowhere to be found. As we were about to continue our hike, I looked into the pond one more time and there it was, the elusive beaver was swimming in the pond, actually from our side of the pond toward its house - the beaver's dam! This was our first ever sighting of beavers in their natural habitat.

Wildlife is not limited by the administrative boundary of the park. On our way to Jackson Hole Mountains, we crossed road again with hard to see wildlife - a mother moose and her 1 day old baby (per a ranger who was monitoring the moose at the scene) were in a pond by the road. The innocent baby was only several yards away from us, it looked at our direction curiously until its mom called. We witnessed the feeding process as well as moose peeing!

For more common animals, such as deers, elks, bison/buffalo on the ground, and pelicans, gees, ravens, gulls in the sky or by the water, all you need to do is to be at the right places. On the beavers pond trail, we met a group of three couples of senior citizens, who told us they had not seen buffaloes - the common big mammals in the park! If you don't know where to see specific animals, ask a park ranger.

There are many buffaloes in the park in the center east area - we saw hundreds of buffaloes at one valley, we saw tens of buffaloes crossing a road at fisherman's pier by Yellowstone lake.

Elks are very common as well, they might rest in a meadow by a road in the early morning, they might walk in a river at noon time, or they might eat grass in late afternoon in a group.

Looking for wildlife is thrilling. Finding some on your own is exciting.

Yellowstone: Water and Mountains

Water of various forms represents a very important facet of Yellowstone area - numerous waterfalls, rivers and streams, ponds and lakes, and most famous of all - geysers, the steaming water.

Shortly after we entered the park from the south, we arrived at the Moose falls. Not many people stop there, we had the whole area for ourselves for 30 minutes or so, listening to the sound of rushing water, feeling the breeze, enjoying the nature. Of all the water falls at Yellowstone, the most impressive one is the lower falls on Yellowstone river, which has a height of 300feet. The water was rushing so fast and forceful, it sent mists into the air forming a light fog. As we went down to the bottom of the fall via Uncle Tom's trail, we saw rainbows!

In addition to Yellowstone river, the larger rivers include the Firehole river which originated from geyser basin. The snake river runs through Yellowstone, and the most beautiful scenery along the river is at its Grand Teton section. We had white water rafting the first day at Grand Teton National Park.

The biggest lake is Yellowstone lake. It is huge and the water is crystal clear. It is so huge we decided to drive a motor boat instead of our usual favorite kayaking. At one corner of the Yellowstone lake, there is a large marsh land - which is said to be bears favorite place, but we did not see any bears there. There are so many ponds we did not even try to remember their names, except the beavers pond - because a trail we hiked is named beavers pond trail and we actually saw beaver's dam and a beaver swimming in the pond just for minute or two when we hiked there.

The most famous form of water there is geyser. After touring the geyser basin, we could appreciate how amazing it is that Old Faithful erupts regularly. Among the less famous geysers, this long horn like dormant geyser caught my eyes. Geysers are amazing, but they smell big time.

The pictures listed below are 1) old faithful- one of its smaller eruption I caught on camera, 2) a dormant geyser, 3) steamboat geyser - what shown in the picture is not a eruption, it was a continuous steam from the boiling geyser, 4) water seeping out of rocks from one side of Calderon geyser

The mountains at Yellowstone are not high. They, along with blue sky, white clouds, remnant of glaciers, lakes, trees and green vegetation, form a beautiful tapestry of Yellowstone.

Bear Encounter at Yellowstone

Watching grizzly bears in the wild was one thing we looked forward to during our 7 day stay at the Yellowstone National Park two years ago. We hiked on the trails which had good chances of bear sighting, we went to marshes that bears usuaully hang around ...despite many attempts, we did not catch even a glimpse of the bears in the first 6 days there.

The last day, we had a fantastic morning - driving a motor boat in the Yellowstone lake, first ever for all of us. In the afternoon we went to Cascade Lake Trail for an afternoon hike, and also our last chance of bear sighting.

It was a breezy, beautiful sunny afternoon. The blue sky was spotted with snow white clouds. After about a quarter mile in the woods, we walked into open Meadows full of wild flowers - white, purple, red and yellow, crystal clear water running in shallow creeks. There were few people there. When we reached cascade lake, we were all by ourselves as far as eye could see, with the company of two hawks flying above the valley.

After snack by the lake, we were on the return hike. As we entered the wooded area, I sensed something moving near a stump by the side of the trail ahead of us. We stopped, the rest of the group saw nothing. "False alarm! You want to see bears too much." Lily teased me.

A few seconds later, a dark brown animal was in sight. "Grizzly bear!" The boys exclaimed, starting to run to the bear. "Stop!" I shouted out in a suppressed voice. "Be careful!" Lily's voice was a bit shaky. We stopped there, watching the bear and took a lot of pictures. The bear apparently did not notice us, snorting around, occasionally raising its head to smell the air, walking slowly toward the lake direction. We stood there watching the bear until it disappeared into the meadow. What a sight! Wow!

We were lucky.

On the way back to htotel, Lily spotted something white in a pond we drove by. We decided to turn around to check it out - We were thrilled to see a trumpeter swan. The lucky day just got luckier for us.

Exploration made it so much fun.