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Topic: A power plus a prime (Read 502 times) |
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NickH
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A power plus a prime
« on: Apr 12th, 2003, 6:00am » |
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Let n, r, and p be positive integers, with p prime.
Show that, for each r > 1, there are infinitely many positive integers that cannot be represented in the form nr + p.
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NickH
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Re: A power plus a prime
« Reply #1 on: Apr 26th, 2003, 12:29pm » |
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Here are two clues...
1) Solve first for r = 2.
2) Consider numbers of the form am + b, for m = 1, 2, ... , with a, b to be determined.
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ThudnBlunder
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Re: A power plus a prime
« Reply #2 on: Apr 26th, 2003, 1:41pm » |
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Quote:Let n, r, and p be positive integers, with p prime.
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This opening line suggests that n, r, and p are fixed (although they 'obviously' are not).
Perhaps you mean:
Prove that there are infinitely many integers S, such that S cannot be represented in the form nr + p for any prime p >= 2 and positive integer values of n and r, (r > 1).
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| « Last Edit: Apr 26th, 2003, 1:44pm by ThudnBlunder » |
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Eigenray
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Re: A power plus a prime
« Reply #3 on: Apr 26th, 2003, 8:23pm » |
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Suppose r > 1 is fixed, and that all but finitely many integers k can be written in the form k = n^r + p, for n positive, p prime.
Then there is some N such that k has the desired form whenever k > N. Then, whenever m > M = N^(1/r), we can find n and p such that nr + p = mr, which means
p = mr - nr = (m-n)(mr-1 + mr-2 n + . . . + mnr-2 + nr-1)
is prime, so that we are forced to choose n = m-1, so that for m > M, we must have the polynomial
f(m) = mr - (m-1)r
take on only prime values. But there can be no such polynomial.
For f(M+1) = p must be prime, and f(M+1+kp) = qk is prime for each k. But then kp | (qk - p), so p | qk, which means f(M+1+kp) = p for all k, which is clearly impossible.
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| « Last Edit: Apr 27th, 2003, 4:42pm by Eigenray » |
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NickH
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Re: A power plus a prime
« Reply #4 on: May 1st, 2003, 3:11pm » |
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Sorry for the misunderstanding I seem to have caused here! Let me try to restate this puzzle more clearly.
Show that there are infinitely many positive integers that cannot be expressed in the form n2 + p, where n and p are positive integers, with p prime.
Generalize to inexpressible in the form n3 + p, and nk + p, for each k > 3.
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