## Monday, February 8, 2016

### primes and quadratic involving deduction of 2

Take a prime p and double it then deduct 5 to obtain 2p5

candidate = -2+p*2p5

Is candidate prime?

Example p=53 so 2p5 =101

5351=-2+53*101
5351 is prime

Next try p=47 so 2p5 = 89

4181=-2+47*89
4181=37*113

Next try p=59 so 2p5 = 113

6665=-2+59*113
6665=5*31*43

Next try p=61 so 2p5 = 117

7135=-2+61*117
7135=5*1427

Next try 67 so 2p5 = 129

8641 is prime

Now algebraically we might say -2+p*(2p-5) which is rewritten 2p^2-5p-2

Next try 71 so 2p5 = 137

9725=-2+71*137
9725=5*5*389

Next try 73 so 2p5 = 141

10291=-2+73*141
10291=41*251

Next try 79 so 2p5 = 153

12085=-2+79*153
12085=5*2417

Next try 83 so 2p5 = 161

13361=-2+83*161
13361=31*431

Next try 89 so 2p5 = 173

15395=-2+89*173
15395=5*3079

Next try 97 so 2p5 = 189

18331=-2+97*189
18331=23*797

Next try 101 so 2p5 = 197

=-2+101*197
=5*23*173

Next try 103 so 2p5 = 201

20701=-2+103*201
20701=127*163 is composite

Next try 107 so 2p5 = 209

22361=-2+107*209
22361=59*379 is composite

Next try 109 so 2p5 = 213

23215=-2+109*213
23215=5*4643 is 5 divisible

Next try 113 so 2p5 = 221

24971=-2+113*221
24971 is prime

Answer will be divisible by 5 when -2+2p^2 is divisible by 5
( or written another way answer cannot be divisible by 5 when -2+2p^2 is not divisible by 5 )

Now we redo the above example but with slightly different assignment for p

Set the prime 113 then deduct a 100 and use that figure as p

13 is prime
p=13

113=p+10^2

221 = p*(p+4)

-2+(p+(2*5)^2)*p*(p+4)

prime 24971==-2+113*221            ( 221 is 13*17 )

p=17

-2+(p+(2*5)^2)*p*(p+4)

41767=-2+117*17*21
41767=11*3797

p=19

-2+(p+(2*5)^2)*p*(p+4)
51201=-2+(7*17)*(19*23)
51201=149*349

p=1053 so p+100 is 1053
p2=1053

prime 1283313211==-2+1153*(1053*1057)

( 1283313213==(3^4)*7*13*151*1153 )

Now let us vary things slightly by defining p2=p+100 and having our answer derived as -2+(p2^2)*(p2+4)

p=953 which is prime and then p+100 is 1053
p2=1053

prime 1172011111==-2+1053*(1053*1057)

( 1172011113==(3^8)*7*(13^2)*151 )