What is the difference between mathematical true proposition and theorem? Why can't a true proposition be used as a theorem?

A theorem is a truth obtained through a lot of application and proof. A true proposition is only a correct proposition, not a theorem. Therefore, since it is a theorem, it can be directly used for reference, just like axiom. Axiom is a truth recognized by all, and can be directly used for reference without proof. For example, 1 + 1 = 2, if you want to prove the reason of 1 + 1 = 2, That's too difficult - the difficulty in mathematical conjecture; so true propositions can't be used as theorems

Reciprocal proposition and reciprocal theorem

Reciprocal proposition
In two propositions, if the proposition of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the proposition of the second proposition, then the two propositions are called reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition
The truth and falseness of original proposition and inverse proposition
Every proposition has an inverse proposition, but the original proposition is a true proposition, and its inverse proposition is not necessarily a true proposition. There are four kinds of truth and falsehood of the original proposition and the inverse proposition: true, false; true, true; false, false; false, true
Reciprocal theorem
If the inverse proposition of a theorem is proved to be true, then it is also a theorem. These two theorems are called reciprocal theorems, and one of them is called the inverse theorem of the other
Every proposition has an inverse proposition, but not all theorems have an inverse theorem

3x^4 +4x^3+7x^2+4x +3
Detailed process!

Factorization
2x^4+x^3+7x^2+4x-4
=2x^4+x^3-x^2+8x^2+4x-4
=x^2(2x^2+x-1)+4(2x^2+x-1)
=(x^2+4)(2x^2+x-1)
=(x^2+4)(2x-1)(x+1)

(1.5 * 1.25 + 1.75 * 1.5) * 5 simple operation
(1.5 * 1.25 + 1.75 * 1.5) * 5 is a simple operation,

(1.5*1.25+1.75*1.5)*5
=1.5*(1.25+1.75)*5
=1.5*3*5
=4.5*5
=22.5

The sequence {an} is an equal ratio sequence, the number of items is even, the items are positive, and the sum of all its items is equal to 4 times of the sum of even items,
And the product of the second term and the fourth term is 9 times of the sum of the third term and the fourth term?

Let the common ratio of the equal ratio sequence = q, the number of terms = 2n, and n be n positive,
If the even term of the sequence {an} is an equal ratio sequence with a1q as the first term and the square of Q as the common ratio, and the sequence has n terms, then
A1 (2n power of 1-Q) / (1-Q) = 4 * a1q [1 - (n power of Q Square)] / (1-Q Square)
Because a is not equal to 0, so
(2n power of 1-Q) / (1-Q) = 4q (2n power of 1-Q) / [(1 + Q) (1-Q)]
Well organized
1+q=4q
q=1/3
From known to known
A1q * A1 (cubic power of Q) = 9 [A1 (square of Q) + A1 (cubic power of Q)]
A1 (square of Q) = 9 (1 + Q)
The solution
a1=108
When an is greater than 1, the sum of sequence is the largest
lgan=lg(a1+a2+a3+a4+a5)=lg(108+36+12+4+4/3)
That is, the first five terms and the maximum of the sequence {lgan}

-24x-3 (20-x) = - 4 this is the first 4 (2 / 1x-2) + 3x = 5-6 (1-2 / 3x) second 1-3 (8-x) = - 2 (15-2x) third
3（x-1）-2（2x+3）=6

-24x-3（20-x）=-4
-24x-60+3x=-4
-21x=56
x=-8/3
4（2/1x-2）+3x=5-6（1-2/3x）
4(2x-2)+3x=5-6(1-2/3x)
8x-8+3x=5-6+4x
7x=7
x=1
1-3（8-x）=-2（15-2x）
1-24+3x=-30+4x
-x=-7
x=7
3（x-1）-2（2x+3）=6
3x-3-4x-6=6
-x=15
x=-15

If for the rational number a, B defines the operation * as follows: a * b = (a-b) - (a-b), then find the value of 3 * 4 * 5

No matter what a and B are, a * b = (a-b) - (a-b) = 0?
3 * 4 * 5 is 0. Is the title copied wrong?

Many four digit numbers can be made up of 1, 2, 3 and 4. If they are arranged from small to large, which number is 4123

There are six at the beginning of 1, six at the beginning of 2, and six at the beginning of 3
4123 is the smallest of 4000 +
So it's the 19th

A is the m * n matrix, B is the n * m matrix. It is proved that R (E-Ab) + n = R (e-Ba) + M

Consider the equation (E-Ab) x = 0, X is an m-dimensional vector, let the dimension of the solution space V of the equation be K, then k = M-R (E-Ab)
Let X be the solution of the equation, then ABX = ex = X. then BA (BX) = B (ABX) = B (x) = (BX), let y = BX, where BA (y) = y, i.e
Y is the solution of the equation (e-Ba) y = 0. Let W be the solution space of the equation
If y belongs to W, with bay = y and x = ay, then BX = y and ab (x) = a (BX) = a (y) = x, that is, X belongs to v
That is to say, B: V - > W is a full homomorphism from V to W, and a: W - > V is a full homomorphism from w to V, so V and W are isomorphic, so their dimensions are equal
So m - R (E-Ab) = n - R (e-Ba)

If a two digit number is divided by a one digit number, the quotient is still two digits, the remainder is 8, and the sum of divisor, divisor and quotient is______ ．

According to the meaning of the question, the minimum divisor is 8 + 1 = 9, and because the divisor is a one digit number, the divisor must be 9, because it is a two digit number divided by a one digit number, and the quotient is still two digits. If the quotient is 10, then: 10 × 9 + 8 = 98, which is in line with the meaning of the question; if the quotient is 11, then: 11 × 9 + 8 = 107, which is not in line with the meaning of the question; so the divisor is 98, the quotient is 10, the divisor is 9, and the remainder is 8, 98 + 10 + 9 + 8 = 125 The sum of divisor, divisor and quotient is 125

What is the difference between mathematical true proposition and theorem? Why can't a true proposition be used as a theorem?