Is a^2+c Always a Prime Number Under Certain Conditions?

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The discussion centers on whether the expression a^2 + c can yield a prime number under specific conditions, such as a being even and c being odd, or vice versa, while ensuring neither is a multiple of the same number and c is not a negative square. Participants argue that simply showing a number is odd does not guarantee it is prime, as demonstrated with examples where a^2 + c results in both prime and non-prime outcomes. The concept of prime numbers being a unique combination of powers of two is questioned, with clarifications that all integers can be represented in binary form, which does not restrict prime numbers. A proposed condition that a^2 - c should not be divisible by a - c is also examined, but fails to consistently yield prime results. Ultimately, the discussion concludes that the initial assertions do not sufficiently establish a reliable method for generating prime numbers.
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dear sir, i wish to know if i am correct. a^2+c can be a prime number provided if a is even then c is odd or vice versa, also a and c are not multiple of same number. and c is not a negative square of any number. finally prime number is unique combination of 1,2,and other powers of 2. each power of two is used only once. i wish to know if prime number is bound by it
thank you
 
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I don't think you have shown more than those are odd numbers. Showing that a number can be prime, because it's odd, is not terribly useful. For a = 3 and b = 2, you getg 11, which is prime. For a = 5 and b = 2 you get 27, which is not.

I don't understand your last two sentences.
 
shivakumar06 said:
finally prime number is unique combination of 1,2,and other powers of 2
Every integer can be expressed this way. It's the binary or 2-adic representation. (I assume you are talking about integers, although it is not quite clear to me, since you just say "numbers" and "negative squares".)
each power of two is used only once. i wish to know if prime number is bound by it
thank you
If you mean "once and only once" by "only once" it is wrong for ##1+2+4+8 = 15##.
If you mean "at most once" by "only once" it is wrong since all integers have such a representation.
So the final answer to your question seems to be: No.
 
Vanadium 50 said:
I don't think you have shown more than those are odd numbers. Showing that a number can be prime, because it's odd, is not terribly useful. For a = 3 and b = 2, you getg 11, which is prime. For a = 5 and b = 2 you get 27, which is not.

I don't understand your last two sentences.
sir if we add a condition that a^2-c should not be divisible a-c. will this satisfy condition for prime number?
 
What about ##a = 12## , ##c = 25## ? ##a## is even, ##c## is odd, they don 't have a common divisor, ## a - c = -13 ## does not divide ## a^2 -c = 144 - 25 = 119 = 7 * 17## and ##a^2 + c = 144 + 25 = 13^2## is not prime.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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