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
If f(x)=3x-2 and g(x)= x^2+1, which expression is equivalent to f(g(x))
Serhud [2]
The answer would be:
3x^2+1
7 0
3 years ago
what is the rate of change of the linear function that has a graph that passes through the points (2,9) and (-1,3)?​
Liono4ka [1.6K]

Answer:

2

Step-by-step explanation:

slope is 2 which is the rate of change

4 0
3 years ago
What is 1000x1000 is please tell me I'll give you Brainly
Eduardwww [97]

Answer:

1,000,000

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Triangle ABC is congruent to triangle DEF. If DE = 41, EF = 23, DF = 36, and AC = 5x - 9, what is x? Round to the nearest tenth.
Aloiza [94]

Similar shapes may or may not be congruent.

The value of x is (a) 9.0

The corresponding sides are:

  • AB and DE
  • BC and EF
  • AC and DF

The side lengths are:

\mathbf{DE = 41}

\mathbf{EF = 23}

\mathbf{DF = 36}

\mathbf{AC = 5x - 9}

Because the shapes are congruent, and AC corresponds to DF.

This means that:

\mathbf{AC = DF}

So, we have:

\mathbf{5x - 9 = 36}

Collect like terms

\mathbf{5x = 9 + 36}

\mathbf{5x = 45}

Divide both sides by 5

\mathbf{x = 9.0}

<em>Hence, the correct option is (a) 9.0</em>

Read more about congruent triangles at:

brainly.com/question/12413243

5 0
2 years ago
Help Please please please please !
il63 [147K]

Step-by-step explanation:

b.

The sum of two opposite interior angles equals one exterior angle

111°=60°+‹G

‹G=111°-60°=51°

c.

We use the same property above

3x+4x+5=68°

7x=68-5

7x=63

x=63/7=9

‹D=4(9)+5

‹D=41°

6 0
3 years ago
Other questions:
  • Find two consecutive integers whose sum is 67. Which of the following equations could be used to solve the problem?
    8·1 answer
  • Find the area?????????
    9·2 answers
  • Can someone answer this question please please help me I really need it if it’s correct I will mark you brainliest .
    15·2 answers
  • A palm tree casts a 24 foot shadow. The end of the shadow is 44 feet from the top of the tree. How tall is the tree? Pls help
    12·1 answer
  • (15 points, will mark brainliest) The sum of two numbers is 56. The smaller number is 14 less than the larger number. What are t
    11·2 answers
  • A wholesale distributor of packaged nuts sells two different mixes of peanuts and cashews, mix A and mix B. Each package of mix
    9·1 answer
  • PLEASE HELP <br> Here is an equation.<br> P = d(4 + tg)<br> What is the equation when solved for g?
    13·2 answers
  • 1. 0.000527 = 5.27 x 10-4<br><br> a. The number 0.000527 is a very<br><br> number.
    11·1 answer
  • Passes through (1,4) and is parallel to the line y=2x-3
    14·1 answer
  • How do answer this please help help me
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!