1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Montano1993 [528]
3 years ago
10

Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square .

Mathematics
1 answer:
kogti [31]3 years ago
4 0

Answer:

The family of all prime numbers such that a^{2} + b^{2} + c^{2} -1 is a perfect square is represented by the following solution:

a is an arbitrary prime number. (1)

b = \sqrt{1 + 2\cdot a \cdot c} (2)

c is another arbitrary prime number. (3)

Step-by-step explanation:

From Algebra we know that a second order polynomial is a perfect square if and only if (x+y)^{2} = x^{2} + 2\cdot x\cdot y  + y^{2}. From statement, we must fulfill the following identity:

a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}

By Associative and Commutative properties, we can reorganize the expression as follows:

a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2} (1)

Then, we have the following system of equations:

x = a (2)

(b^{2}-1) = 2\cdot x\cdot y (3)

y = c (4)

By (2) and (4) in (3), we have the following expression:

(b^{2} - 1) = 2\cdot a \cdot c

b^{2} = 1 + 2\cdot a \cdot c

b = \sqrt{1 + 2\cdot a\cdot c}

From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, a, b, c > 1. If a, b and c are prime numbers, then  2\cdot a\cdot c must be an even composite number, which means that a and c can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.

In addition, b must be a natural number, which means that:

1 + 2\cdot a\cdot c \ge 4

2\cdot a \cdot c \ge 3

a\cdot c \ge \frac{3}{2}

But the lowest possible product made by two prime numbers is 2^{2} = 4. Hence, a\cdot c \ge 4.

The family of all prime numbers such that a^{2} + b^{2} + c^{2} -1 is a perfect square is represented by the following solution:

a is an arbitrary prime number. (1)

b = \sqrt{1 + 2\cdot a \cdot c} (2)

c is another arbitrary prime number. (3)

Example: a = 2, c = 2

b = \sqrt{1 + 2\cdot (2)\cdot (2)}

b = 3

You might be interested in
PLS Help! What is the best approximation for the perimeter of a semicircle with a diameter of 12cm? Use 3.14 to approximation pi
In-s [12.5K]

Answer:

A

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
Error Analysis Your friend says that the volume of this sphere is 44.58
Maurinko [17]

Answer:

  • the given dimension was used as the radius
  • 5.57 m³

Step-by-step explanation:

The volume of a sphere can be found using the formula ...

  V = 4/3πr³ . . . . . where r is the radius

__

The figure points to a diameter line and indicates 2.2 m. The arrowhead is in the middle of a radius line, making it easy to interpret the dimension as the radius of the sphere.

If 2.2 m is used as the radius, the volume is computed to be ...

  V = 4/3π(2.2 m)³ ≈ 44.58 m³

This agrees with your friend's volume, suggesting the diameter was used in place of the radius in the computation.

__

The correct volume, using 2.2 m as the diameter, is ...

  V = 4/3π(1.1 m)³ ≈ 5.57 m³

6 0
2 years ago
What is 3123532488697 x 5676576876896786
horrorfan [7]
The answer to ur question is 1.77309723E28.
Hope this helps and can i get brainliest answer.
4 0
3 years ago
Read 2 more answers
Find two integers whose sum is -20 and product is 99
Brums [2.3K]

Answer:

Step-by-step explanation:

let the integers be x and y

x+y=-20

y=-20-x

xy=99

x(-20-x)=99

-20x-x²=99

x²+20x+99=0

x²+11x+9x+99=0

x(x+11)+9(x+11)=0

(x+11)(x+9)=0

x=-11,-9

when x=-11

-11+y=-20

y=-20=11=-9

when x=-9

-9+y=-20

y=-20+9=-11

so integers  are -11,-9

8 0
2 years ago
Read 2 more answers
What is the product (2x + 3y) (7x – 4y)?
cupoosta [38]

Answer:

14x²-12y²

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Other questions:
  • In a candy mix there are 4 green bars for every 3 red bars how many red bars are ther if there are 200 green bars
    11·2 answers
  • Ginger's cat gave birth to a kitten that weighed 3 3/8 ounces when it was born. On the day Ginger sold the kitten to its new own
    11·2 answers
  • What is the answer to 11 pls help me!
    6·1 answer
  • Determine the constant of Proportionality in the situation below
    9·1 answer
  • The isosceles trapezoids, ABCD and EFGH, are similar quadrilaterals. The scale factor between the trapezoids is 2:3, = 6 centime
    11·1 answer
  • Given the following sets.
    9·2 answers
  • When I divide both sides of an inequality by a negative number, I notice...
    9·2 answers
  • -ive
    9·1 answer
  • Solve the equation by finding the common multiple. (1 point)
    15·2 answers
  • JUST DO IT PKEASE DIWOSJJDNKA
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!