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Mayank Kotwala
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Today, 6-28, is a Perfect Day
June 28, or 6-28, is a “perfect day” because both 6 and 28 are so-called perfect numbers.

A perfect number is one whose divisors add up to itself. The divisors of 6 are 1, 2, and 3, and 1+2+3=6. Likewise, 28=1+2+4+7+14. After these two, the next three perfect numbers are 496, 8128, and 33550336.

2,300 years ago, Euclid discovered a way to generate perfect numbers. First, we find a number n such that 2ⁿ – 1 is a prime number. Then, that prime multiplied by 2ⁿ⁻¹ is a perfect number. For example, n = 3 gives 2³ – 1 = 7, which is prime. So, 7 x 2² = 28 is a perfect number. Such perfect numbers are always even.

In 1755, Leonard Euler proved that there can be no even perfect numbers except for those that Euclid’s formula generates. However, he could not determine whether there any perfect numbers that are odd. 261 years later, this question is still unresolved.


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The Harmonic Series is Infinite
Most of the infinite series that we meet have a finite sum. For example, in high school mathematics, we learn that 1 + 1/2 + 1/4 + 1/8 + 1/16 + … has a sum of 2 (geometric series).

This seems possible – though there are an infinite number of terms, the terms become infinitely small.

However, a surprise awaits with the series 1 + 1/2 + 1/3 + 1/4 + … Called the harmonic series, this sum can be proven to be infinite, even though the terms are becoming infinitely small. Roughly speaking, the terms do not become small fast enough.

The fact that the sum is infinite was proven in the year 1350 by Nicole Oresme. This proof is included in a list of the 100 greatest mathematics theorems, and it uses only basic arithmetic.

In his proof, shown in the graphic, he writes another infinite series which has two properties. First, its sum is larger than that of the harmonic series. Second, the new series has an infinite sum because it is equivalent to an infinite numbers of terms, each with a value of 1/2. Therefore the harmonic series is also infinite.



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Mental Math – Squaring a Number Near 100

1. Subtract 100 from the number. Call this d.
2. Add d to the original number.
3. Append d ²

Example: 109²
1. d = 109 - 100 = 9
2. 109 + 9 = 118
3. d ² = 81, so answer = 11881

Example: 92²
1. 92 - 100 = -8
2. 92 + (-8) = 84
3. d ² = 64, so answer = 8464

Obviously, this works only for numbers in the range 91 to 109.

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There Are Infinitely Many Prime Numbers
Euclid’s famous proof of this is shown here.

It’s a proof by contradiction: assume there are only a finite number of prime numbers, and we have the list of them. Euclid shows that we can always use that list to find another prime that is not in the original list. This contradicts the original assumption, so that the alternative must be true: there are an infinitude of prime numbers.

Mathematician G. H. Hardy said “This proof is as fresh and significant as when it was discovered–two thousand years have not written a wrinkle on it”



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NGC 3628: The Hamburger Galaxy
Explanation: No, hamburgers are not this big. What is pictured is a sharp telescopic view of a magnificent edge-on spiral galaxy NGC 3628, a puffy galactic disk divided by dark dust lanes. Of course, this deep galactic portrait puts some astronomers in mind of its popular moniker, The Hamburger Galaxy. The tantalizing island universe is about 100,000 light-years across and 35 million light-years away in the northern springtime constellation Leo. NGC 3628shares its neighborhood in the local Universe with two other large spirals M65 and M66 in a grouping otherwise known as the Leo Triplet. Gravitational interactions with its cosmic neighbors are likely responsible for the extended flare and warp of this spiral's disk.
Image Credit & Copyright: Eric Coles and Mel Helm
#nasa #esa #spaceexploration

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