Prime numbers are like atoms. They are the building blocks of all integers. Every integer is either itself a prime or the product of primes. For example, 11 is a prime; 12 is the product of the primes 2, 2, and 3; 13 is a prime; 14 is the product of the primes 2 and 7; 15 is the product of the primes 3 and 5; and so on. Some 2,300 years ago, in proposition 20 of Book IX of his Elements, Euclid gave a proof “straight from the book” that the supply of primes is inexhaustible.
Assume, said Euclid, that there is a finite number of primes. Then one of them, call it P, will be the largest. Now consider the number Q, larger than P, that is equal to the product of the consecutive whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 … x P) + 1.
From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into it; each division would leave a remainder of 1. If Q is not prime, it must be evenly divisible by some prime larger than P. On the other hand, if Q is prime, Q itself is a prime larger than P.
Either possibility implies the existence of a prime larger than the assumed largest prime P. This means, of course, that the concept of “the largest prime” is a fiction. But if there’s no such beast, the number of primes must be infinite. “Euclid alone,” wrote Edna St. Vincent Millay, “has looked upon Beauty bare.”
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Paul Hoffman (and Euclid) in The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth.
if you don’t think this is the coolest fucking thing idek what to say
For the uninitiated, the James Webb Space Telescope (JWST) - due to launch in 2018 - is a 6.5 meter telescope which will peer 13.5 billion years into the early universe, revealing the meticulous formation and cosmic evolution of stars and galaxies. JWST’s infrared capability will permit insight into the birth of planetary and stellar systems within the densely opaque interstellar dust clouds which visible-light observatories - such as the Hubble Space Telescope (HST) - cannot provide.
The JWST will aid in the search for life in the universe by analyzing the atmospheres of extrasolar planets or, planetary bodies orbiting stars outside our solar system. Understanding the atmospheric chemical composition of other atmospheres will guide our roughly 13.8-14.5 billion year quest to find the building blocks of life in the universe.
An international collaboration between the National Aeronautics and Space Administration (NASA), European Space Agency (ESA), and Canadian Space Agency (CSA), the JWST will be managed by NASA’s Goddard Space Flight Center (GSFC), Northrop Grumman, and after launch, the Space Telescope Science Institute (STScI), which also currently operates the Hubble Space Telescope as a part of John Hopkins University.
The “James Webb” in James Webb Space Telescope serves as a respectful ode to NASA’s former second administrator - James E. Webb - an influential proponent for space science throughout and beyond the Apollo program. When chosen to be administrator for NASA, Webb is quoted as saying, “I’m not going to run a program that’s just a one-shot program. If you want me to be the administrator, it’s going to be a balanced program that does the job for the country….”
As NASA Administrator Sean O’Keefe said when he announced the new name for the next generation space telescope, “It is fitting that Hubble’s successor be named in honor of James Webb. Thanks to his efforts, we got our first glimpses at the dramatic landscape of outer space. He took our nation on its first voyages of exploration, turning our imagination into reality. Indeed, he laid the foundations at NASA for one of the most successful periods of astronomical discovery. As a result, we’re rewriting the textbooks today with the help of the Hubble Space Telescope , the Chandra X-ray Observatory , and the James Webb Telescope.”
So what’s with the obsessive hexagonal construction inside honeybees’ hives? And why is this honeycomb structure of relative importance to the design of the James Webb Space Telescope and its functionality? NPR Science Correspondent Robert Krulwich explains via the NPR blogpost
While at the 30th Space Symposium, I had the privilege of enjoying a conference session with those responsible for James Webb’s creation, implementation, and ultimately, it’s success and discovery: Blake Bullock, Director, Civil Air and Space, Business & Advanced Systems Development - Northrop Grumman Aerospace Systems; Dave Gallagher, Director for Astronomy, Physics and Space Technology - NASA Jet Propulsion Laboratory; John M. Grunsfeld, Associate Administrator for the Science Mission Directorate - NASA; Matt Mountain, Director, Space Telescope Science Institute; John C. Mather, Senior Project Scientist, James Webb Space Telescope - NASA; and Sara Seager, Professor of Planetary Science and Physics - Massachusetts Institute of Technology (MIT). It was by far one of the most inspiring discussions regarding the future I’ve experienced thus far.
Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations: x = A.sin(a.t + δ) and y = B.cos(bt) The appearance of the figure is highly sensitive to the ratio a/b - Image 3 (3/2, ¾ and 5/4). For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/4). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.
Visually, the ratio a/b determines the number of “lobes” of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes (see image). The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high. Finally, the value of δ determines the apparent “rotation” angle of the figure, viewed as if it were actually a three-dimensional curve. For example, δ=0 produces x and y components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left/right or an up/down rotation (depending on the ratio a/b).
A reservoir of water three times the volume of all the oceans has been discovered deep beneath the Earth’s surface. The finding could help explain where Earth’s seas came from. The water is hidden inside a blue rock called ringwoodite that lies 700 kilometres underground in the mantle, the layer of hot rock between Earth’s surface and its core. The huge size of the reservoir throws new light on the origin of Earth’s water. Some geologists think water arrived in comets as they struck the planet, but the new discovery supports an alternative idea that the oceans gradually oozed out of the interior of the early Earth. “It’s good evidence the Earth’s water came from within,” says Steven Jacobsen of Northwestern University in Evanston, Illinois. The hidden water could also act as a buffer for the oceans on the surface, explaining why they have stayed the same size for millions of years.
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