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kaheart [24]
3 years ago
14

If 2^n + 1 is an odd prime for some integer n, prove that n is a power of 2. (H

Mathematics
1 answer:
vovikov84 [41]3 years ago
4 0

Step-by-step explanation:

We will prove by contradiction. Assume that 2^n + 1 is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that p\mid n. Then, for some integer k\geq 1,

n=p\times k.

Therefore

  1. 2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p.

Here we will use the formula for the sum of odd powers, which states that, for a,b\in \mathbb{R} and an odd positive number n,

a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})

Applying this formula in 1) we obtain that

2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1).

Then, as 2^k+1>1 we have that 2^n+1 is not a prime number, which is a contradiction.

In conclusion, if 2^n+1 is an odd prime, then n must be a power of 2.

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Let x = 10000

0.2^(10000) + 3 = 3.00032

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Answer:

  ∠B ≈ 30.0°

Step-by-step explanation:

The law of sines can be used to solve a triangle when two sides and an angle opposite one of them are given.

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<em>Additional comments</em>

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A triangle has side lengths that are 8 cm, 15 cm, 17 cm. Is this a right triangle?
Alik [6]

Answer: A triangle has side lengths of 8 cm, 15 cm, and 17 cm. classify it as acute, obtuse, or right. Well, since 8^2 + 15^2 = 17^2, the sides fit the Pythagorean Theorem and thus it must be a right triangle.

Step-by-step explanation:

A triangle has side lengths that are 8 cm, 15 cm, 17 cm. Is this a right triangle? A triangle has side lengths of 8 cm, 15 cm, and 17 cm. classify it as acute, obtuse, or right. Well, since 8^2 + 15^2 = 17^2, the sides fit the Pythagorean Theorem and thus it must be a right triangle.

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3 years ago
Find the volume of the solid whose base is the region Ix I Iy I &lt; 1 and whose vertical cross sections perpendicular to the y-
gavmur [86]

Answer:

volume of the solid = \frac{\pi }{3}

Step-by-step explanation:

As given , the region : |x| + |y| \leq 1

when y \geq 0 ,

|x| + y = 1

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when y ≤ 0 ,

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The given region is the rectangle with sides (0, 5) , (0, -5), (5, 0), (-5, 0)

As given , the vertical cross section are semicircle

As we know that the area of semi circle  = \frac{1}{2}\pi  x^{2}

Now,

Volume = \int\limits^0_-1    {\frac{\pi }{2} (1+y)^{2}  } \, dy + \int\limits^1_0    {\frac{\pi }{2} (1-y)^{2}  } \, dy

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⇒volume of the solid = \frac{\pi }{3}

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