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Do all natural numbers exist on these three lines?
The Next CEO of Stack OverflowHow to find the sum of Big-Oh's?Proof with $Theta$Prove that even + odd is odd.Is this a valid proof of the squeeze theorem?Proof: Fibonacci Sequence (2 parts)3 different proofsFor any natural numbers a, b, c, d if a*b = c*d is it possible that a + b + c + d is prime numberFor all natural numbers $n,; n^2-n$ is even.Proof that Cardinality of Primes is Cardinality of Natural NumbersShow that no ∞-persistent numbers exist.
$begingroup$
The lines given by
$$f_1:mathbbWrightarrow mathbbN,~~f_1(n)=(6n+1)cdot2^k$$
$$f_2:mathbbWrightarrow mathbbN,~~f_2(n)=(6n+3)cdot2^k$$
$$f_3:mathbbWrightarrow mathbbN,~~f_3(n)=(6n+5)cdot2^k$$
where $n,k$ are arbitrary whole numbers.
By ignoring the $2^k$ I can see that any given odd number will appear on one of the lines, and when multiplied by a factor of 2 will give any even number. How do you suppose I would go about and prove all of the natural numbers exist on the lines?
proof-writing
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
$endgroup$
add a comment |
$begingroup$
The lines given by
$$f_1:mathbbWrightarrow mathbbN,~~f_1(n)=(6n+1)cdot2^k$$
$$f_2:mathbbWrightarrow mathbbN,~~f_2(n)=(6n+3)cdot2^k$$
$$f_3:mathbbWrightarrow mathbbN,~~f_3(n)=(6n+5)cdot2^k$$
where $n,k$ are arbitrary whole numbers.
By ignoring the $2^k$ I can see that any given odd number will appear on one of the lines, and when multiplied by a factor of 2 will give any even number. How do you suppose I would go about and prove all of the natural numbers exist on the lines?
proof-writing
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
$endgroup$
add a comment |
$begingroup$
The lines given by
$$f_1:mathbbWrightarrow mathbbN,~~f_1(n)=(6n+1)cdot2^k$$
$$f_2:mathbbWrightarrow mathbbN,~~f_2(n)=(6n+3)cdot2^k$$
$$f_3:mathbbWrightarrow mathbbN,~~f_3(n)=(6n+5)cdot2^k$$
where $n,k$ are arbitrary whole numbers.
By ignoring the $2^k$ I can see that any given odd number will appear on one of the lines, and when multiplied by a factor of 2 will give any even number. How do you suppose I would go about and prove all of the natural numbers exist on the lines?
proof-writing
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
$endgroup$
The lines given by
$$f_1:mathbbWrightarrow mathbbN,~~f_1(n)=(6n+1)cdot2^k$$
$$f_2:mathbbWrightarrow mathbbN,~~f_2(n)=(6n+3)cdot2^k$$
$$f_3:mathbbWrightarrow mathbbN,~~f_3(n)=(6n+5)cdot2^k$$
where $n,k$ are arbitrary whole numbers.
By ignoring the $2^k$ I can see that any given odd number will appear on one of the lines, and when multiplied by a factor of 2 will give any even number. How do you suppose I would go about and prove all of the natural numbers exist on the lines?
proof-writing
proof-writing
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
asked Mar 19 at 22:18
Alex MaslachAlex Maslach
275
275
add a comment |
add a comment |
1 Answer
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$begingroup$
Your observation is pretty much correct, you just need to formalise it. You can start by proving that every odd $x$ is in one of the sequences, probably by assuming that there's an $x$ not in $f_1$ or $f_2$, and hence showing that it must be in $f_3$. After that, if you can then prove that any even number can be written as $2^kcdot x$ for some odd $x$, you're done.
answered Mar 19 at 22:24
ConManConMan
7,9671324
$endgroup$
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1 Answer
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1 Answer
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$begingroup$
Your observation is pretty much correct, you just need to formalise it. You can start by proving that every odd $x$ is in one of the sequences, probably by assuming that there's an $x$ not in $f_1$ or $f_2$, and hence showing that it must be in $f_3$. After that, if you can then prove that any even number can be written as $2^kcdot x$ for some odd $x$, you're done.
answered Mar 19 at 22:24
ConManConMan
7,9671324
$endgroup$
add a comment |
$begingroup$
Your observation is pretty much correct, you just need to formalise it. You can start by proving that every odd $x$ is in one of the sequences, probably by assuming that there's an $x$ not in $f_1$ or $f_2$, and hence showing that it must be in $f_3$. After that, if you can then prove that any even number can be written as $2^kcdot x$ for some odd $x$, you're done.
answered Mar 19 at 22:24
ConManConMan
7,9671324
$endgroup$
add a comment |
$begingroup$
Your observation is pretty much correct, you just need to formalise it. You can start by proving that every odd $x$ is in one of the sequences, probably by assuming that there's an $x$ not in $f_1$ or $f_2$, and hence showing that it must be in $f_3$. After that, if you can then prove that any even number can be written as $2^kcdot x$ for some odd $x$, you're done.
answered Mar 19 at 22:24
ConManConMan
7,9671324
$endgroup$
Your observation is pretty much correct, you just need to formalise it. You can start by proving that every odd $x$ is in one of the sequences, probably by assuming that there's an $x$ not in $f_1$ or $f_2$, and hence showing that it must be in $f_3$. After that, if you can then prove that any even number can be written as $2^kcdot x$ for some odd $x$, you're done.
answered Mar 19 at 22:24
ConManConMan
7,9671324
answered Mar 19 at 22:24
ConManConMan
7,9671324
answered Mar 19 at 22:24
ConManConMan
7,9671324
answered Mar 19 at 22:24
ConManConMan
7,9671324
7,9671324
add a comment |
add a comment |
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