## A valid Observation by a program on Goldbach's conjecture up to 10^6.

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• REM Goldbach s conjecture is finally prooved by induction, the full analytic deductive proof should be one that explains the following inductive proof. The
Message 1 of 1 , Jan 25, 2005
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REM Goldbach's conjecture is finally prooved by induction, the
full analytic deductive proof should be one that explains the
following inductive proof. The inductive proof states that for every
2n>2 : (P_i)^2 <= 2n < (P_[i+1])^2 ,were P_ i is the ith prime
like P_3 = 5 , at least one of the 2n- P_1 or 2n-P_2 or 2n-P_3
or........or 2n-P_2i should be prime! Since every even number has
primes below it more than or equal to 2i , then ALL 2n fulfills this
inductive proof and Goldbach's conjecture is true for all 2n>2. This
was induced from the fact that ratio of all primes below any even
number 2n>2 : (P_i)^2 <= 2n < (P_[i+1])^2 to the number of
building primes of the composite numbers up to that 2n ( which is
of cours i )is larger or equal to 2, and it increases as 2n increase.
REM Now a deductive full proof should explain why ? I think
overlapes of the building primes like 15+ 30j and 21 + 42j and 35 +
70j and....etc , are important key numbers,since it is in the
vicinity of these numbers there is a room for 2n-P_f to be prime.
REM the inductive proof was tested by this program up to 10^6, it
was true for all evens <= 10^6 and >2. And I assume that it is true
for all evens>2.

3 z = 0
5 INPUT x
8 INPUT s
REM s shouldn't be more than the square of the last number in the
data list.
10 INPUT v
15 INPUT b
REM it is v and b that shows' the infinite nature of this program,
as they are decreased the number of prime numbers subtracted from y
would be less , and this program shows' that as y increase,
Goldbach's conjecture becomes true at less values of v and b.thus
supporting views writtin in the message titled raw ideas?. For the
purpose of validating the inductive statement outlined above both v
and b should equal 1.
20 FOR y = x TO s STEP 2
g = 0
30 k = 1
40 w = 0
50 FOR I = 1 TO k

70 n = y - P

90 NEXT I
100 RESTORE 260
110 FOR j = 1 TO 172
130 IF n = q THEN 180
140 IF (n / q) = INT(n / q) THEN 190
150 IF n <= q ^ 2 THEN 180
160 NEXT j
170 GOTO 190
180 w = w + 1
185 GOTO 235
190 RESTORE 260
200 IF v * y < P ^ 2 THEN 230
210 k = k + 1
220 GOTO 50
230 g = g + 1
231 IF g <> 1 THEN 233
232 o = INT(b * k)
233 IF g = o THEN 235
234 GOTO 210

235 RESTORE 260
237 IF w <> 0 THEN 250
z = z + 1
240 PRINT y; z;
250 NEXT y
260 DATA 3,5,7,
11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,10
3,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193
,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,
293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,4
01,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,50
3,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617
,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,
739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,8
57,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,97
7,983,991,997,1009,1013,1019,1021

263 IF z <> 0 THEN 270

265 PRINT "Goldbach's conjecture is true!!!"
270 IF s = 0 THEN 300
280 GOTO 3

300 END

REM I hope the above information would help.
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