_{1}

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We prove that e ＋ π is a transcendental number. We use proof by contradiction. The key to solve the problem is to establish a function that doesn’t satisfy the relational expression that we derive, thereby pr oduce a conflicting result which can verify our assumption is incorrect.

Hilbert’s seventh problem is about transcendental number. The proof of transcendental number is not very easy. We have proved the transcendence of “e” and “π”. However, for over a hundred years, no one can prove the transcendence of “e + π” [

1) Assuming

Now we consider this integral:

2) Assuming

According to Formula (2.1), using

So, all we need to do or the key to solve the problem is to find a suitable

3) So we let

Furthermore, we consider

By the analysis above, we can know that

Now we see

and its the sum of the first p − 1 item is zero (because the degree of each term of

, (2.3)

4) Next, we need to prove that

When x changes from 0 to n, the absolute value of each factor

So by integral property: when

Let M equal

thus,

When

Finally, according to (2.3) and (2.4), we know (2.2) is incorrect. So, e + π is a transcendental number.

By the proof above, we conclude that e + π is a transcendental number. Besides, I suppose

I am grateful to my friends and my classmates for supporting and encouraging me.

Jiaming Zhu, (2016) The Proof of Hilbert’s Seventh Problem about Transcendence of e＋π. Open Access Library Journal,03,1-3. doi: 10.4236/oalib.1102893