Maths Primes and Multiples Investigation
. A)I) ?(3)-1, 2=2
II) ?(8)-1, 2, 3, 4, 5, 6, 7=4
III) ?(11)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10=10
IV) ?(24)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23=8
B)I) ?(5)-1, 2, 3, 4=4
II) ?(10)- 1, 2, 3, 4, 5, 6, 7, 8, 9=4
III) ?(15)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14=8
IV) ?(20)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19=8
V) ?(50)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49=20
2. A)I) ?(7x4) = ?(7) x ?(4)
7x4=28
?(28)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27=12
?(7)=1, 2, 3, 4, 5, 6=6
?(4)=1, 2, 3=2
2x6=12
12=12, therefore a prime and an even work(non-prime).
B) ?(6x4) = ?(6) x ?(4)
6x4=24
?(24)=8
?(6)=1, 2, 3, 4, 5=2
?(4)=2
2x2=4
4=8, therefore two evens don't work.
C) ?(5x10)= ?(50)
?(50)=20
?(10) x ?(5)
4x4=16
16=20, therefore two multiples don't work.
?(13x3)= ?(39)
?(39)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38=24
?(13)=12
?(3)=2
2x12=24
24=24, therefore two primes work.
Having done these examples I below have drawn up a table of my results. When I made the table I saw that there are lots of other combinations that I need to investigate.
Prime
Non Prime
Odd
Even
Multiple
Prime
Yes
Yes
Non Prime
Odd
Even
No
Multiple
No
3. To complete my table and to test out this idea I will try a variety of numbers in different combinations. For example I will be using odds, evens and prime numbers in different combinations. For each cell in my table I will do two different examples to make sure that it is the same for all numbers. I will start by doing a prime and a non prime number.
?(3x4) = ?(3) x ?(4)
?(3)=2
?(4)=2
2x2=4
?(12)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11=4
4=4, therefore a prime and a non-prime work sometimes.
To finish my non-prime column I will try two non-primes now:
?(4x6) = ?(4) x ?(6)
?(6)=2
?(4)=2
2x2=4
?(24)=8
4=8, therefore two non-primes don't work.
I will try another example just to make sure that this is correct:
?(9x4) = ?(4) x ?(9)
?(9)=1, 2, 3, 4, 5, 6, 7, 8=6
?(4)=2
6x2=12
?(36)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35=12
12=12, therefore two non-primes sometimes work.
I will now try an even with one of its multiples and an odd with one of its multiples to help complete my table.
?(4x8) = ?(8) x ?(4)
?(8)=4
?(4)=2
4x2=8
?(32)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31=16
16=8, therefore an even and one of its multiple doesn't work.
?(3x9) = ?(3) x ?(9)
?(3)=2
?(9)=1, 2, 3, 4, 5, 6, 7, 8=6
6x2=12
?(27)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26=18
12=18, therefore an odd and one of its multiple doesn't work.
When I filled these values into my table I saw that the odd and even ...
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21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31=16
16=8, therefore an even and one of its multiple doesn't work.
?(3x9) = ?(3) x ?(9)
?(3)=2
?(9)=1, 2, 3, 4, 5, 6, 7, 8=6
6x2=12
?(27)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26=18
12=18, therefore an odd and one of its multiple doesn't work.
When I filled these values into my table I saw that the odd and even I used were also prime and non-prime respectively. I can therefore also fill in those two boxes in the table. I also see that some of the other boxes can be filled in for the same reason. Nine and four are an even and an odd so I can fill that in. I will try an even and a non-prime now, as I have no good examples of this set of numbers.
?(8x4) = ?(8) x ?(4)
?(8)=4
?(4)=2
4x2=8
?(32)=16
8=16, therefore an even and a non-prime don't work.
As I found anomalies with non-primes before I will now check this is correct by doing another one with the same type of numbers.
?(8x9) = ?(8) x ?(9)
?(8)=4
?(9)=6
4x6=24
?(72)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71=24
24=24, therefore an even and a non-prime work sometimes.
To complete my table I need to complete the odd column, that I will do now. However I saw that I had already done an example for an odd and a prime, three and nine, so I will test this by doing another one:
?(5x9) = ?(5) x ?(9)
?(5)=4
?(9)=6
4x6=24
?(45)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44=24
24=24, therefore an odd and a prime work sometimes.
I have already done eight and nine, ten and five so I think that I have done enough examples to say that odds and non-primes work only sometimes.
I have also already done thirteen and three, three and nine and I think that I have done enough examples to say that two odds also work only sometimes.
Here is a table of all the phis that I have done so that I can refer to them so I don't do any more than once.
Number
Phi
3
2
4
2
5
4
6
2
7
6
8
4
9
6
0
4
1
0
2
4
3
2
5
8
20
8
24
8
27
8
28
2
32
6
36
2
39
24
50
20
72
24
From this table I can easily see that larger numbers have bigger phis. There is only this simple correlation though; there is no exact relationship. I will now put my completed table to see if it gives me any other relationships.
Here is my completed table with the new data:
Prime
Non Prime
Odd
Even
Multiple
Prime
Yes
Yes/No
Yes/No
Yes
No
Non Prime
Yes/No
Yes/No
Yes/No
No
Odd
Yes/No
Yes
No
Even
No
No
Multiple
No
Now that I have filled in my table I will look at it to see if it shows me why some numbers work and some don't. Looking at the table, I can see that primes work with anything except their multiples, and some numbers work only some of the time. I cannot seem to make much else out of this table so I will try arranging it in a different format:
st number
2nd Number
Works?
Even
Odd
Yes
Odd
Odd
Yes
Even
Odd
Yes
Even
Even
No
Even
Even
No
Even
Odd
No
Odd
Odd
No
No numbers work all the time here. The only numbers that don't work all the time are two evens. Now I will try a different format of table to see if it shows me any more correlations.
st number
2nd Number
Works?
Prime
Non-prime
Yes
Prime
Prime
Yes
Prime
Non-prime
Yes
Prime
Non-prime
No
Non-prime
Non-prime
No
Prime
Non-prime
No
Prime
Non-prime
No
The only two numbers that work all the time are two primes.
The two numbers that always work are two primes. They have no multiples
The two numbers that never work are too evens. All evens have the at least on multiple, because two goes into all evens. Therefore two evens all have the same lowest common factor.
I will now look at some of the numbers that sometimes work to see why they only work on some occasions. I will use the example of two non-primes:
4 and 6 doesn't work.
4 and 9 does work.
Four and six are both evens and therefore have the same common factor. 4 and 9 don't have the same lowest common factor, that's why evens and odd never work. I will check this with the second example of an even and a non-prime:
4 and 8 doesn't work
8 and 9 does work.
Four and eight are again both evens and therefore have the same common factor of two. Eight and nine don't have the same lowest common factor, and they therefore don't work, odds and even never work.
From these two examples I can see that two numbers with the same common factor don't work. I can therefore do an example that I have not done before to test this, ten and eleven. I predict that ten and eleven work, as they don't have a common factor.
?(5x6) = ?(5) x ?(6)
?(5)=4
?(6)=2
4x2=8
?(30)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29=8
8=8, therefore my theory works.
4. A) I have done five primes number already, 3, 5, 7, 11 and 13. Prime numbers will always be odd numbers because the number two always goes into even numbers. I will do some more and then put them in a table to compare them.
?(17)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16=16
?(19)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18=18
?(23)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22=22
?(29)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28=28
?(31)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30=30
Number
Phi
3
2
5
4
7
6
1
0
3
2
7
6
9
8
23
22
29
28
31
30
Looking at this table I can see that the phi of a prime number is one less than the number itself. This is because no numbers go into primes and therefore the phi is the addition all of the numbers. I can therefore write this formula:
?Prime = (n-1)
B) I already have one n2 number, (3 and 9). I need some more examples, and will do them, then put them all in a table.
?(52)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24=20
?(72)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48=42
I have now got enough examples of p2 and will put my findings into a table to try and find any relationships.
Prime
Phi1
Prime2
Phi2
3
2
9
6
5
4
25
20
7
6
49
42
I can see from this table that the phi of the prime2 is always more than the phi of the prime. I will now do another table in which I will add and subtract various numbers from the table above. I am doing this to try and find any patterns I can't see on this table. I will be looking for any calculations that will result in the phi of the prime2.
Numbers
Prime + ?1
Prime2 + ?1
Prime2 - ?1
Prime2 - Prime
3, 9
5
1
7
6
5, 25
9
29
21
20
7, 49
3
55
43
42
Correlations?
No
No
No
Yes
Looking at this table I found a correlation. I noticed that if you minus the prime from the prime2, you get the phi of the prime2. Now that I know this, I can write this formula:
?Prime2 = Prime2 - Prime
Using this formula I can now predict other values. Take 13 for example. Using the formula I can work out the phi of thirteen squared:
Prime
Phi1
Prime2
Phi2
3
2
69
56
I worked out the phi of one hundred and sixty nine to be one hundred and fifty six.
C) I will start this investigation by creating a table. This is because I already know some of the values for three, five and seven as I have done them before. I will now make a simple table, and any values that I work out I will type in red.
p
?(p1)
?(p2)
?(p3)
?(p4)
3
2
6
8
54
5
4
20
00
500
7
6
42
84
2058
3
2
56
2028
26364
I knew that the phi of 132 was 156 from part B). Also from part B) I knew that p2=p x (p-1). When I looked at the column when p=3, I saw that p2 x (p-1) = ?(p3). I used this formula to work out the rest of the missing values in the column of ?(p3). Using these two formulas I realised that ?(p4)=p3 x (p-1). I used this formula to work out the column of values for ?(p4). I found my formulas it on the following theory:
?(p2) = p x (p-1)
?(p3) = p x p x (p-1)
?(p4) = p x p x p x (p-1)
Using all of these formulas that I have found I worked out one for ?(pn). It is:
?(pn) = pn-1 x (p-1)
Using this formula I can predict the formula of any prime to the power of any number (except to the power of 1). Take 17 for example:
?(173) = 172 x (17-1)
?(173) = 289 x 16
?(4913) = 289 x 16
?(4913) = 4624
I can therefore say that this formula is correct and will work with any prime number to the power of n.
D) This is the most complicated part of my investigation. I have noticed that this question involves parts of the whole of D). It is:
?(pnqm)
This means the phi of (p to the n time q to the m), when p and q are prime. When I looked at the expression closely I realised that it was similar to the equation in Part 2. In Part 2 I was finding out whether the phi of two numbers multiplied together was the same as the two numbers phis multiplied. The equation was:
?(n x m) = ?(n) x ?(m)
I found out that this equation didn't work if the two numbers had the same lowest common factor. The two numbers I will be using for this section will both be primes. Therefore their multiples will not have a lowest common factor, and they would always work using this formula. For example, if I used 3 and 5,
?(32 x 52)
?(225)=120
? (9)=6
? (25)=20
6 x 20=120
20=120, so the powers of two primes work, as they don't have a common factor
As I have shown, the powers of primes do not have a common factor. I can therefore put the formula for this section into an easier form:
?(pn) x ?(qm)
Having the formula in this form will be easier because I will not have to work out the phis of very large numbers to confirm my findings. I can use some of the data that I found in section C) to help me find the phis of some numbers. I can now start to do some more numbers to try and build up a picture using the formula ?(pn) x ?(qm):
? (33)=18
? (52)=20
18 x 20=360
? (34)=54
? (52)=20
54 x 20=1080
? (32)=2
? (53)=100
2 x 100=200
? (32)=2
? (54)=500
2 x 500=1000
? (52)=20
? (72)=42
20 x 42=840
? (53)=100
? (72)=42
100 x 42=42,000
? (54)=500
? (72)=42
42 x 500=21,000
? (52)=20
? (73)=84
20 x 84=1680
? (52)=20
? (74)=2058
20 x 2058=41,160
I will now out all of my numbers in a table to try and compare them:
st Number
Power
2nd Number
Power
Phi
3
2
5
2
20
3
3
5
2
360
3
4
5
2
,080
3
2
5
3
200
3
2
5
4
,000
5
2
7
2
840
5
3
7
2
42,000
5
4
7
2
21,000
5
2
7
3
680
5
2
7
4
41,160
The Phi Function
