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#76 2005-07-13 19:58:02
- MathsIsFun
- Administrator
- Registered: 2005-01-21
- Posts: 7,713
Re: I need your help
It just struck me that every such proof usually starts with the 1, 2 example ... and I just assumed it was necessary.
But really if you prove that it moves from "n" to "n+1" that should be enough - it is proven all up and down the natural number scale, including 1.
(I think!)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#77 2005-07-13 19:59:01
- MathsIsFun
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Re: I need your help
MathsIsFun wrote:Where did I say that?
Here. Fifth post down.
Ahh... so it was YOU that got me to admit to that LOL
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#78 2005-07-14 05:52:29
- Maxine
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- Registered: 2005-07-14
- Posts: 3
Re: I need your help
Wow, you discovered a new number pattern thingy!
sorry because i dident understand ur story
Maxine
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#79 2005-07-14 17:57:42
- Jai Ganesh
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- Registered: 2005-06-28
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Re: I need your help
I discovered that 2^20 ends in 76.
Thereafter, any number of the form 2^20n ends in 76.
Similarly, any number of the form 2^100n ends in 376.
Next, I had to know the last few digits of 2^500 and 2^2500.
When I learnt that they are 9376 and 09376, I was excited.
Because, any number of the form 2^500n would then have to end in 9376
and every 2^2500n would have to end in 09376.
This is what Rora was talking about. You are too young to understand this, Maxine. This is about higher powers of the number 2.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#80 2005-08-11 16:10:13
- Jai Ganesh
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- Registered: 2005-06-28
- Posts: 48,413
Re: I need your help
If Mx is the Least Common Multiple of all numbers from 1 to x,
can it be proved that
x^x < 2^Mx < x^x^x ?
The second part is very simple. 2^Mx is certainly less than x^x^x because both the base and the exponent of 2^Mx are lesser than x^x^x.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#81 2005-08-11 20:59:02
- mathsyperson
- Moderator
- Registered: 2005-06-22
- Posts: 4,900
Re: I need your help
I hate to break it to you, but 2^Mx > x^x^x for x = 0 or 1.
Why did the vector cross the road?
It wanted to be normal.
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#82 2005-08-11 21:17:49
- Jai Ganesh
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- Registered: 2005-06-28
- Posts: 48,413
Re: I need your help
0^0^0 is not defined.
But for 1, as you pointed out, 2^Mx > x^x^x.
For this problem, let us take value of x ≥ 2, x ∈ N
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#83 2005-08-11 21:42:30
- MathsIsFun
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- Registered: 2005-01-21
- Posts: 7,713
Re: I need your help
Is this something you are working on, ganesh? Something you want our help with?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#84 2005-08-11 22:20:56
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 48,413
Re: I need your help
You are right, MathsisFun. I am working on this. I shall give you the details next week.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#85 2005-08-29 03:40:56
- John E. Franklin
- Guest
Re: I need your help
Answer to original question at top was correct according to "Just BASIC" program I wrote:
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
11 2048
12 4096
13 8192
14 16384
15 32768
16 65536
17 31072
18 62144
19 24288
20 48576
21 97152
22 94304
23 88608
24 77216
25 54432
26 8864
27 17728
28 35456
29 70912
30 41824
31 83648
32 67296
33 34592
34 69184
35 38368
36 76736
37 53472
38 6944
39 13888
40 27776
41 55552
42 11104
43 22208
44 44416
45 88832
46 77664
47 55328
48 10656
49 21312
50 42624
51 85248
52 70496
53 40992
54 81984
55 63968
56 27936
57 55872
58 11744
59 23488
60 46976
61 93952
62 87904
63 75808
64 51616
65 3232
66 6464
67 12928
68 25856
69 51712
70 3424
71 6848
72 13696
73 27392
74 54784
75 9568
76 19136
77 38272
78 76544
79 53088
80 6176
81 12352
82 24704
83 49408
84 98816
85 97632
86 95264
87 90528
88 81056
89 62112
90 24224
91 48448
92 96896
93 93792
94 87584
95 75168
96 50336
97 672
98 1344
99 2688
100 5376
101 10752
102 21504
103 43008
104 86016
105 72032
106 44064
107 88128
108 76256
109 52512
110 5024
111 10048
112 20096
113 40192
114 80384
115 60768
116 21536
117 43072
118 86144
119 72288
120 44576
121 89152
122 78304
123 56608
124 13216
125 26432
126 52864
127 5728
128 11456
129 22912
130 45824
131 91648
132 83296
133 66592
134 33184
135 66368
136 32736
137 65472
138 30944
139 61888
140 23776
141 47552
142 95104
143 90208
144 80416
145 60832
146 21664
147 43328
148 86656
149 73312
150 46624
151 93248
152 86496
153 72992
154 45984
155 91968
156 83936
157 67872
158 35744
159 71488
160 42976
161 85952
162 71904
163 43808
164 87616
165 75232
166 50464
167 928
168 1856
169 3712
170 7424
171 14848
172 29696
173 59392
174 18784
175 37568
176 75136
177 50272
178 544
179 1088
180 2176
181 4352
182 8704
183 17408
184 34816
185 69632
186 39264
187 78528
188 57056
189 14112
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191 56448
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220 40576
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228 87456
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231 99648
232 99296
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234 97184
235 94368
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242 79104
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511 42048
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513 68192
514 36384
515 72768
516 45536
517 91072
518 82144
519 64288
520 28576
521 57152
522 14304
523 28608
524 57216
525 14432
526 28864
527 57728
528 15456
529 30912
530 61824
531 23648
532 47296
533 94592
534 89184
535 78368
536 56736
537 13472
538 26944
539 53888
540 7776
541 15552
542 31104
543 62208
544 24416
545 48832
546 97664
547 95328
548 90656
549 81312
550 62624
551 25248
552 50496
553 992
554 1984
555 3968
556 7936
557 15872
558 31744
559 63488
560 26976
561 53952
562 7904
563 15808
564 31616
565 63232
566 26464
567 52928
568 5856
569 11712
570 23424
571 46848
572 93696
573 87392
574 74784
575 49568
576 99136
577 98272
578 96544
579 93088
580 86176
581 72352
582 44704
583 89408
584 78816
585 57632
586 15264
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591 88448
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597 60672
598 21344
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600 85376
601 70752
602 41504
603 83008
604 66016
605 32032
606 64064
607 28128
608 56256
609 12512
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611 50048
612 96
613 192
614 384
615 768
616 1536
617 3072
618 6144
619 12288
620 24576
621 49152
622 98304
623 96608
624 93216
625 86432
626 72864
627 45728
628 91456
629 82912
630 65824
631 31648
632 63296
633 26592
634 53184
635 6368
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637 25472
638 50944
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696 86336
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699 90688
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701 62752
702 25504
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715 28768
716 57536
717 15072
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719 60288
720 20576
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727 33728
728 67456
729 34912
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768 57856
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770 31424
771 62848
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781 56352
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791 4448
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802 9504
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807 4128
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813 64192
814 28384
815 56768
816 13536
817 27072
818 54144
819 8288
820 16576
821 33152
822 66304
823 32608
824 65216
825 30432
826 60864
827 21728
828 43456
829 86912
830 73824
831 47648
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833 90592
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971 78848
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978 92544
979 85088
980 70176
981 40352
982 80704
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990 60224
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992 40896
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996 54336
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1001 38752
1002 77504
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1016 25536
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Sorry about the disk space I must be using somewhere...
#86 2005-09-28 01:47:39
- mathsyperson
- Moderator
- Registered: 2005-06-22
- Posts: 4,900
Re: I need your help
How very stretchy of you.
Why did the vector cross the road?
It wanted to be normal.
Offline
#87 2005-11-15 00:34:56
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 48,413
Re: I need your help
I'm not good at proofs, but this is my attempt:
Take 109376 as an example.
We need to prove that (y109376)² ends in y109376 for only one value of y.
y109376 can be written as y*10^6+1*10^5+0*10^4+9*10^3+3*10^2+7*10^1+6*10^0.
We are only interested in the 10^6 coefficient of the square of this. That will be equal to 2(y*(6*10^0))+a constant made up of the expansion of the other powers of 10, but that does not involve y.
So, the 10^6 coefficient is 12y+c, which should only be equal to y for one value of y.
We are only interested in the last digit of 12y+c, so it can be called 2y+c without affecting our purposes. However big c is, it can also have all of its digits except the last removed.
We want to solve y=2y+c.
Take away y+c: y=-c
Give back one of c's 10's to make it more sensible: y=10-c
c is a constant, so there is only one value for y.As shown in the expansion of y109376, the important coefficient will always take the form 2(y*6), because 6 is always the last digit. This means that this proof can be carried forward for any point on the chain of the magic number.[/proof]
There's probably a flaw in there somewhere, could someone check it please?
Thanks a million, Mathsyperson! I have checked the proof thoroughly and now, I can declare the proof is precise, elegant, and flawless.
This proof has been very helpful to me.
I thank you, again, MathsIsFun, for the Full Precision Calculator.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#88 2005-11-15 04:42:31
- mathsyperson
- Moderator
- Registered: 2005-06-22
- Posts: 4,900
Re: I need your help
Woo! For the number that ends in 5, the partner to the one you found, if you apply my proof to that then you will eventually get that the last digit is the last digit of 10y+c, and so would also be the last digit of c. It's much easier to continue the chain of the number ending in 5 because of that property and, as mentioned earlier, the one ending in 6 adds to the one ending in 5 to give 1000...0001, so you can use that to work out the one ending in 6 much quicker.
Why did the vector cross the road?
It wanted to be normal.
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#89 2005-11-16 04:10:59
- Alisha
- Guest
Re: I need your help
I must be really bad at moth cause I have no clue what you guys are talking about!
#90 2005-11-16 18:59:08
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 48,413
Re: I need your help
It may not be true that you are bad at Math, Alisha...the topic we're discussing is not taught at School...simple as that 
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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