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August 5th, 2015, 07:27 AM   #1
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Conjecture about odd prime numbers

Every odd prime number p can be expressed at least once as :

p=(2^a + or - 2^b)/k where

a and b integers > 0
a+b < p
k some integer > 0

Examples :

5=2^2+2^0
7=2^3-2^0
11=(2^5+2^0)/3
23=(2^11-2^0)/89

Any counterexample?
Any proof?
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August 5th, 2015, 08:23 AM   #2
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You say that a and be must be > 0, but then use examples with b = 0. What's the rule supposed to be?
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August 5th, 2015, 08:37 AM   #3
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Proof: Consider the p-1 residues of 2^1, 2^2, ..., 2^(p-1) mod p. If any two, say a and b, yield the same residues, then 2^a - 2^b is a proper multiple of p. If not, then each residue class from 1 to p-1 is represented (0 can't be since p is odd). Then just take, say, the values which yield residues of 1 and p-1 and add them together.

This proves more: in fact, we can require that 0 < b < a <= (p+3)/2, I believe.

Edit: I missed the restriction a+b < p. This isn't a big problem, but the above proof doesn't cover it.
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August 5th, 2015, 09:01 AM   #4
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p=1783
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August 5th, 2015, 10:07 AM   #5
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Quote:
Originally Posted by mobel View Post
p=1783
1783 = (2^892 - 2^1) / 18518456643846930957387120895839757659802556296835 44314806368161198380196847745581444255664216409099 46254344851814753462872387675092079899484905699124 69839347601888810957833853362714444630764198437473 11091598388965746032994782535539955120929497463812 7087022605781618.
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August 8th, 2015, 06:49 AM   #6
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Quote:
Originally Posted by mobel View Post
Every odd prime number p can be expressed at least once as :

p=(2^a + or - 2^b)/k where

a and b integers > 0
a+b < p
k some integer > 0

Examples :

5=2^2+2^0
7=2^3-2^0
11=(2^5+2^0)/3
23=(2^11-2^0)/89

Any counterexample?
Any proof?
Sorry I did a mistake.
a and b >=0
I forget the sign =
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August 8th, 2015, 06:54 AM   #7
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Can we express the same way as expressed above any odd composite number?

n=15 yes (2^4-2^0)/1
n=21 yes (2^6-2^0)/3

I do not know for others.

Last edited by mobel; August 8th, 2015 at 06:58 AM.
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August 8th, 2015, 07:04 AM   #8
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Quote:
Originally Posted by mobel View Post
Can we express the same way as expressed above any odd composite number?
You'll notice that my proof above doesn't use the fact that the number is prime, so it holds for odd composites as well.
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August 8th, 2015, 10:02 AM   #9
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Formally, we come to diofantos equation.

$\displaystyle pk=2^a\pm2^b$

This equation is solvable for any simple $\displaystyle p$

Because a different number can be set to any value. Because any composite number can be represented as the product of different primes.
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August 8th, 2015, 07:50 PM   #10
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This is near (if not equivalent) to a conjecture of my own I posted here years ago.

Let $p,q$ be odd prime numbers, then for any $p$ we find $q$ and a non zero natural $n$ such that $p=q\pm 2^n$. Notice that this conjecture looks stronger, in the sense that the universe of candidates is much more restricted.

I think CGR found a counter example, i.e., both $p=q+2^n$ and $p=q-2^n$ fail for a fixed $p$.

This is not an easy problem. Below 1000 you have

5=3+2^1
7=3+2^2
7=5+2^1
11=3+2^3
11=7+2^2
13=5+2^3
13=11+2^1
17=13+2^2
19=3+2^4
19=11+2^3
19=17+2^1
23=7+2^4
23=19+2^2
29=13+2^4
31=23+2^3
31=29+2^1
37=5+2^5
37=29+2^3
41=37+2^2
43=11+2^5
43=41+2^1
47=31+2^4
47=43+2^2
53=37+2^4
59=43+2^4
61=29+2^5
61=53+2^3
61=59+2^1
67=3+2^6
67=59+2^3
71=7+2^6
71=67+2^2
73=41+2^5
73=71+2^1
79=47+2^5
79=71+2^3
83=19+2^6
83=67+2^4
83=79+2^2
89=73+2^4
97=89+2^3
101=37+2^6
101=97+2^2
103=71+2^5
103=101+2^1
107=43+2^6
107=103+2^2
109=101+2^3
109=107+2^1
113=97+2^4
113=109+2^2
131=3+2^7
131=67+2^6
131=127+2^2
137=73+2^6
139=11+2^7
139=107+2^5
139=131+2^3
139=137+2^1
151=23+2^7
151=149+2^1
157=29+2^7
157=149+2^3
163=131+2^5
167=103+2^6
167=151+2^4
167=163+2^2
173=109+2^6
173=157+2^4
179=163+2^4
181=53+2^7
181=149+2^5
181=173+2^3
181=179+2^1
191=127+2^6
193=191+2^1
197=181+2^4
197=193+2^2
199=71+2^7
199=167+2^5
199=191+2^3
199=197+2^1
211=83+2^7
211=179+2^5
223=191+2^5
227=163+2^6
227=211+2^4
227=223+2^2
229=101+2^7
229=197+2^5
229=227+2^1
233=229+2^2
239=223+2^4
241=113+2^7
241=233+2^3
241=239+2^1
257=193+2^6
257=241+2^4
263=7+2^8
263=199+2^6
269=13+2^8
271=239+2^5
271=263+2^3
271=269+2^1
277=149+2^7
277=269+2^3
281=277+2^2
283=251+2^5
283=281+2^1
293=37+2^8
293=229+2^6
293=277+2^4
307=179+2^7
311=307+2^2
313=281+2^5
313=311+2^1
317=61+2^8
317=313+2^2
347=283+2^6
347=331+2^4
349=317+2^5
349=347+2^1
353=97+2^8
353=337+2^4
353=349+2^2
359=103+2^8
367=239+2^7
367=359+2^3
379=251+2^7
379=347+2^5
383=127+2^8
383=367+2^4
383=379+2^2
389=373+2^4
397=269+2^7
397=389+2^3
401=337+2^6
401=397+2^2
409=281+2^7
409=401+2^3
419=163+2^8
421=293+2^7
421=389+2^5
421=419+2^1
431=367+2^6
433=401+2^5
433=431+2^1
439=311+2^7
439=431+2^3
443=379+2^6
443=439+2^2
449=193+2^8
449=433+2^4
457=449+2^3
461=397+2^6
461=457+2^2
463=431+2^5
463=461+2^1
467=211+2^8
467=463+2^2
479=223+2^8
479=463+2^4
487=359+2^7
487=479+2^3
491=487+2^2
499=467+2^5
499=491+2^3
503=439+2^6
503=487+2^4
503=499+2^2
521=457+2^6
523=11+2^9
523=491+2^5
523=521+2^1
541=29+2^9
541=509+2^5
547=419+2^7
557=541+2^4
563=307+2^8
563=499+2^6
563=547+2^4
569=313+2^8
571=59+2^9
571=443+2^7
571=563+2^3
571=569+2^1
577=449+2^7
577=569+2^3
587=331+2^8
587=523+2^6
587=571+2^4
593=337+2^8
593=577+2^4
601=89+2^9
601=569+2^5
601=593+2^3
601=599+2^1
607=479+2^7
607=599+2^3
613=101+2^9
617=601+2^4
617=613+2^2
619=107+2^9
619=491+2^7
619=587+2^5
619=617+2^1
631=503+2^7
631=599+2^5
641=577+2^6
643=131+2^9
643=641+2^1
647=631+2^4
647=643+2^2
653=397+2^8
659=643+2^4
661=149+2^9
661=653+2^3
661=659+2^1
673=641+2^5
677=421+2^8
677=613+2^6
677=661+2^4
677=673+2^2
683=619+2^6
691=179+2^9
691=563+2^7
691=659+2^5
691=683+2^3
709=197+2^9
709=677+2^5
709=701+2^3
719=463+2^8
727=599+2^7
727=719+2^3
733=701+2^5
739=227+2^9
743=487+2^8
743=727+2^4
743=739+2^2
751=239+2^9
751=719+2^5
751=743+2^3
761=757+2^2
769=257+2^9
769=641+2^7
769=761+2^3
773=709+2^6
773=757+2^4
773=769+2^2
787=659+2^7
797=541+2^8
797=733+2^6
811=683+2^7
811=809+2^1
821=757+2^6
823=311+2^9
823=821+2^1
827=571+2^8
827=811+2^4
827=823+2^2
829=317+2^9
829=701+2^7
829=797+2^5
829=821+2^3
829=827+2^1
839=823+2^4
853=821+2^5
857=601+2^8
857=853+2^2
859=347+2^9
859=827+2^5
859=857+2^1
863=607+2^8
863=859+2^2
881=877+2^2
883=881+2^1
887=631+2^8
887=823+2^6
887=883+2^2
911=907+2^2
919=887+2^5
919=911+2^3
929=673+2^8
937=809+2^7
937=929+2^3
941=877+2^6
941=937+2^2
947=691+2^8
947=883+2^6
953=937+2^4
967=839+2^7
971=907+2^6
971=967+2^2
983=727+2^8
983=919+2^6
983=967+2^4
991=479+2^9
991=863+2^7
991=983+2^3
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